\usepackage or this document’s own macros. Corpus-wide, \bm occurs 327 times, \Perp is defined by no package and spans 10 documents, and 11,088 of 11,624 undefined occurrences are the source document’s own macros. A row that looks wrong here may be correct, and one that looks right here may not compile. The LaTeX report renders through the document’s own preamble and is the surface to judge from.3116 rows
The page, the confidence and the picture of an inline formula are its HOST LINE's --- a formula has none of its own. A line's confidence is not a formula's.
| Identifier | Page | Conf. | LaTeX source | Rendered | Image |
|---|---|---|---|---|---|
| 0902.0431_FO0001 | 1 | 1.000 | A_{n}, B_{n}, C_{n}, D_{n} | | |
| 0902.0431_FO0002 | 1 | 1.000 | G_{2}, F_{4}, E_{6}, E_{7}, E_{8} | | |
| 0902.0431_FO0003 | 1 | 1.000 | G | | |
| 0902.0431_FO0004 | 1 | 1.000 | \sigma | | |
| 0902.0431_FO0005 | 1 | 1.000 | G^{\sigma} | | |
| 0902.0431_FO0006 | 1 | 1.000 | G / G^{\sigma} | | |
| 0902.0431_FO0007 | 7 | 1.000 | G_{2} | | |
| 0902.0431_FO0008 | 6 | 1.000 | \mathfrak{C} | | |
| 0902.0431_FO0009 | 44 | 1.000 | \mathfrak{d}_{4} | | |
| 0902.0431_FO0010 | 8 | 1.000 | \mathfrak{g}_{2} | | |
| 0902.0431_FO0011 | 16 | 1.000 | \mathfrak{s u}(3) | | |
| 0902.0431_FO0012 | 17 | 1.000 | \mathfrak{g}_{2}{ }^{C} | | |
| 0902.0431_FO0013 | 21 | 1.000 | w | | |
| 0902.0431_FO0014 | 23 | 0.942 | S U(3) | | |
| 0902.0431_FO0015 | 21 | 1.000 | \gamma | | |
| 0902.0431_FO0017 | 26 | 1.000 | z\left(G_{2}\right) | | |
| 0902.0431_FO0018 | 26 | 1.000 | G_{2}{ }^{C} | | |
| 0902.0431_FO0019 | 27 | 1.000 | G_{2(2)} | | |
| 0902.0431_FO0020 | 8 | 1.000 | S O(8) | | |
| 0902.0431_FO0021 | 33 | 0.867 | \operatorname{Spin}(7) | | |
| 0902.0431_FO0022 | 33 | 0.996 | \operatorname{Spin}(8) | | |
| 0902.0431_FO0023 | 38 | 1.000 | F_{4} | | |
| 0902.0431_FO0024 | 36 | 1.000 | \mathfrak{J} | | |
| 0902.0431_FO0025 | 41 | 1.000 | \mathfrak{f}_{4} | | |
| 0902.0431_FO0026 | 46 | 1.000 | \mathfrak{f}_{4}{ }^{C} | | |
| 0902.0431_FO0027 | 56 | 0.990 | \operatorname{Spin}(9) | | |
| 0902.0431_FO0028 | 64 | 1.000 | z\left(F_{4}\right) | | |
| 0902.0431_FO0029 | 64 | 0.965 | (S p(1) \times S p(3)) / \boldsymbol{Z}_{2} | | |
| 0902.0431_FO0030 | 68 | 1.000 | (S U(3) \times S U(3)) / \boldsymbol{Z}_{3} | | |
| 0902.0431_FO0031 | 71 | 1.000 | F_{4}{ }^{C} | | |
| 0902.0431_FO0032 | 72 | 1.000 | F_{4(4)} | | |
| 0902.0431_FO0033 | 72 | 1.000 | F_{4(-20)} | | |
| 0902.0431_FO0034 | 73 | 0.999 | E_{6} | | |
| 0902.0431_FO0035 | 73 | 1.000 | \mathfrak{e}_{6} | | |
| 0902.0431_FO0036 | 46 | 1.000 | \mathfrak{e}_{6}{ }^{C} | | |
| 0902.0431_FO0037 | 76 | 1.000 | A \vee B | | |
| 0902.0431_FO0038 | 5 | 1.000 | \tau | | |
| 0902.0431_FO0039 | 90 | 1.000 | z\left(E_{6}\right) | | |
| 0902.0431_FO0040 | 91 | 0.909 | (U(1) \times \operatorname{Spin}(10)) / \boldsymbol{Z}_{4} | | |
| 0902.0431_FO0041 | 95 | 0.935 | (S p(1) \times S U(6)) / \boldsymbol{Z}_{2} | | |
| 0902.0431_FO0042 | 87 | 1.000 | \tau \gamma | | |
| 0902.0431_FO0045 | 71 | 0.993 | S U(3)) / \boldsymbol{Z}_{3} | | |
| 0902.0431_FO0046 | 73 | 0.999 | E_{6}{ }^{C} | | |
| 0902.0431_FO0047 | 110 | 1.000 | E_{6(6)}, E_{6(2)}, E_{6(-14)} | | |
| 0902.0431_FO0048 | 42 | 1.000 | E_{6(-26)} | | |
| 0902.0431_FO0049 | 113 | 0.961 | E_{7} | | |
| 0902.0431_FO0050 | 111 | 1.000 | \mathfrak{P}^{C} | | |
| 0902.0431_FO0051 | 113 | 1.000 | \mathfrak{e}_{7} | | |
| 0902.0431_FO0052 | 113 | 0.881 | \mathfrak{e}_{7}{ }^{C} | | |
| 0902.0431_FO0053 | 94 | 0.999 | U(1) | | |
| 0902.0431_FO0054 | 134 | 1.000 | z\left(E_{7}\right) | | |
| 0902.0431_FO0055 | 129 | 1.000 | \iota | | |
| 0902.0431_FO0057 | 208 | 0.986 | (S U(2) \times S p i n(12)) / \boldsymbol{Z}_{2} | | |
| 0902.0431_FO0059 | 151 | 1.000 | (S U(3) \times S U(6)) / \boldsymbol{Z}_{3} | | |
| 0902.0431_FO0060 | 113 | 0.961 | E_{7}{ }^{C} | | |
| 0902.0431_FO0061 | 160 | 1.000 | E_{7(7)}, E_{7(-5)} | | |
| 0902.0431_FO0062 | 160 | 1.000 | E_{7(-25)} | | |
| 0902.0431_FO0063 | 166 | 0.988 | E_{8} | | |
| 0902.0431_FO0064 | 162 | 1.000 | \mathfrak{e}_{8}{ }^{C} | | |
| 0902.0431_FO0065 | 165 | 1.000 | E_{8}{ }^{C} | | |
| 0902.0431_FO0066 | 177 | 1.000 | v | | |
| 0902.0431_FO0067 | 189 | 0.834 | \left(S U(2) \times E_{7}\right) / \boldsymbol{Z}_{2} | | |
| 0902.0431_FO0068 | 177 | 0.955 | \widetilde{\lambda} \gamma | | |
| 0902.0431_FO0069 | 181 | 0.854 | S s(16) | | |
| 0902.0431_FO0070 | 189 | 1.000 | z\left(E_{8}\right) | | |
| 0902.0431_FO0072 | 177 | 1.000 | w_{3} | | |
| 0902.0431_FO0074 | 177 | 1.000 | z_{5} | | |
| 0902.0431_FO0075 | 197 | 0.999 | (S U(5) \times S U(5)) / \boldsymbol{Z}_{5} | | |
| 0902.0431_FO0076 | 204 | 1.000 | E_{8(8)} | | |
| 0902.0431_FO0077 | 204 | 1.000 | E_{8(-24)} | | |
| 0902.0431_FO0078 | 5 | 1.000 | \boldsymbol{R}, \boldsymbol{C}=\boldsymbol{R} \oplus \boldsymbol{R} e_{1}, \boldsymbol{H}=\boldsymbol{R} \oplus \boldsymbol{R} e_{1} \oplus \boldsymbol{R} e_{2} \oplus \boldsymbol{R} e_{3} | | |
| 0902.0431_FO0079 | 5 | 1.000 | \boldsymbol{R} | | |
| 0902.0431_FO0080 | 5 | 1.000 | V | | |
| 0902.0431_FO0081 | 5 | 1.000 | \{u+i v \mid u, v \in V\} | | |
| 0902.0431_FO0082 | 5 | 1.000 | V^{C} | | |
| 0902.0431_FO0083 | 5 | 1.000 | \boldsymbol{R}^{C} | | |
| 0902.0431_FO0084 | 5 | 1.000 | C | | |
| 0902.0431_FO0085 | 5 | 1.000 | K | | |
| 0902.0431_FO0086 | 5 | 1.000 | V(K=\boldsymbol{R}, \boldsymbol{C}, C), \operatorname{Iso}_{K}(V) | | |
| 0902.0431_FO0087 | 5 | 0.963 | f | | |
| 0902.0431_FO0088 | 5 | 0.963 | V, V_{f} | | |
| 0902.0431_FO0089 | 5 | 0.963 | \{v \in V \mid f(v)=v\} | | |
| 0902.0431_FO0090 | 5 | 1.000 | V, W(K=\boldsymbol{R}, C), \operatorname{Hom}_{K}(V, W) | | |
| 0902.0431_FO0091 | 5 | 0.915 | f: V \rightarrow W . \operatorname{Hom}_{K}(V, V) | | |
| 0902.0431_FO0092 | 5 | 0.915 | \operatorname{Hom}_{K}(V) | | |
| 0902.0431_FO0093 | 5 | 1.000 | \{g \in G \mid \sigma(g)= | | |
| 0902.0431_FO0094 | 5 | 1.000 | g\} | | |
| 0902.0431_FO0095 | 5 | 1.000 | s \in G, G^{s} | | |
| 0902.0431_FO0096 | 5 | 1.000 | \left\{g \in G \mid s g s^{-1}=g\right\} | | |
| 0902.0431_FO0097 | 5 | 1.000 | X, Y, X \simeq Y | | |
| 0902.0431_FO0098 | 5 | 1.000 | X | | |
| 0902.0431_FO0099 | 5 | 1.000 | Y | | |
| 0902.0431_FO0100 | 5 | 1.000 | G, G^{\prime}, G \cong G^{\prime} | | |
| 0902.0431_FO0101 | 5 | 1.000 | G^{\prime} | | |
| 0902.0431_FO0102 | 5 | 1.000 | G, G^{\prime} | | |
| 0902.0431_FO0103 | 5 | 1.000 | G=G^{\prime} | | |
| 0902.0431_FO0104 | 5 | 1.000 | M(n, K) | | |
| 0902.0431_FO0105 | 5 | 1.000 | n \times n | | |
| 0902.0431_FO0106 | 5 | 1.000 | E=\operatorname{diag}(1, \cdots, 1) \in M(n, K) | | |
| 0902.0431_FO0107 | 5 | 1.000 | A \in M(n, K),{ }^{t} A | | |
| 0902.0431_FO0108 | 5 | 1.000 | A | | |
| 0902.0431_FO0109 | 5 | 1.000 | A^{*} | | |
| 0902.0431_FO0110 | 5 | 0.999 | A: A^{*}={ }^{t} \bar{A} | | |
| 0902.0431_FO0111 | 5 | 0.977 | O(n)=\left\{\left.A \in M(n, \boldsymbol{R})\right|^{t} A A=E\right\} | | |
| 0902.0431_FO0112 | 5 | 0.982 | S O(n)=\{A \in O(n) \mid \operatorname{det} A=1\} | | |
| 0902.0431_FO0113 | 5 | 1.000 | U(n)=\left\{A \in M(n, \boldsymbol{C}) \mid A^{*} A=E\right\} | | |
| 0902.0431_FO0114 | 5 | 1.000 | \left\{A \in M(n, C) \mid \tau\left({ }^{t} A\right) A=E\right\} | | |
| 0902.0431_FO0115 | 5 | 0.534 | S U(n)=\{A \in U(n) \mid \operatorname{det} A=1\} \quad | | |
| 0902.0431_FO0116 | 5 | 0.936 | \operatorname{Sp}(n)=\left\{A \in M(n, \boldsymbol{H}) \mid A^{*} A=E\right\} | | |
| 0902.0431_FO0117 | 5 | 0.942 | \mathfrak{g} | | |
| 0902.0431_FO0118 | 5 | 0.942 | \mathfrak{s u}(n) | | |
| 0902.0431_FO0119 | 5 | 0.942 | \operatorname{SU}(n) | | |
| 0902.0431_FO0120 | 6 | 1.000 | \left\{e_{0}=1, e_{1}, e_{2}, e_{3}, e_{4}, e_{5}, e_{6}, e_{7}\right\} | | |
| 0902.0431_FO0121 | 6 | 1.000 | e_{1}, e_{2}, e_{3} | | |
| 0902.0431_FO0122 | 6 | 1.000 | e_{1} e_{6}=e_{7}, e_{4} e_{7}=e_{3} | | |
| 0902.0431_FO0123 | 6 | 1.000 | e_{2}, e_{5}, e_{7} | | |
| 0902.0431_FO0124 | 6 | 1.000 | e_{5} e_{7}=e_{2} . e_{0}=1 | | |
| 0902.0431_FO0125 | 6 | 1.000 | x 1, x \in \boldsymbol{R} | | |
| 0902.0431_FO0126 | 6 | 1.000 | x | | |
| 0902.0431_FO0127 | 6 | 1.000 | \bar{x} | | |
| 0902.0431_FO0128 | 6 | 1.000 | (x, y) | | |
| 0902.0431_FO0129 | 6 | 1.000 | |x| | | |
| 0902.0431_FO0130 | 6 | 1.000 | R(x) | | |
| 0902.0431_FO0131 | 6 | 1.000 | x \in \mathfrak{C}, x \neq 0 | | |
| 0902.0431_FO0132 | 6 | 1.000 | \frac{\bar{x}}{|x|^{2}} | | |
| 0902.0431_FO0133 | 6 | 1.000 | x^{-1} | | |
| 0902.0431_FO0134 | 6 | 1.000 | x x^{-1}=x^{-1} x=1 | | |
| 0902.0431_FO0135 | 6 | 1.000 | x(y z)=(x y) z | | |
| 0902.0431_FO0136 | 6 | 1.000 | x y=y x | | |
| 0902.0431_FO0137 | 7 | 1.000 | a, b, x, y \in \mathfrak{C} | | |
| 0902.0431_FO0138 | 7 | 0.987 | 1 \quad(x y, x y)=(x, x)(y, y), \quad|x y|=|x||y| | | |
| 0902.0431_FO0139 | 7 | 0.644 | 2 \quad(a x, a y)=(a, a)(x, y)=(x a, y a) | | |
| 0902.0431_FO0140 | 7 | 0.998 | 3(a x, b y)+(b x, a y)=2(a, b)(x, y) | | |
| 0902.0431_FO0141 | 7 | 1.000 | 4 \quad(a x, y)=(x, \bar{a} y), \quad(x a, y)=(x, y \bar{a}) | | |
| 0902.0431_FO0142 | 7 | 0.424 | 5 \quad \overline{\bar{x}}=x, \quad \overline{x+y}=\bar{x}+\bar{y}, \quad \overline{x y}=\bar{y} \bar{x} | | |
| 0902.0431_FO0143 | 7 | 0.982 | 6 \quad(x, y)=(y, x)=\frac{1}{2}(\bar{x} y+\bar{y} x)=\frac{1}{2}(x \bar{y}+y \bar{x}), \quad \bar{x} x=x \bar{x}=|x|^{2} | | |
| 0902.0431_FO0144 | 7 | 0.988 | 7 \quad a(\bar{a} x)=(a \bar{a}) x, \quad a(x \bar{a})=(a x) \bar{a}, \quad x(a \bar{a})=(x a) \bar{a} | | |
| 0902.0431_FO0145 | 7 | 1.000 | a(a x)=(a a) x, \quad a(x a)=(a x) a, \quad x(a a)=(x a) a | | |
| 0902.0431_FO0146 | 7 | 1.000 | 8 \bar{b}(a x)+\bar{a}(b x)=2(a, b) x=(x a) \bar{b}+(x b) \bar{a} | | |
| 0902.0431_FO0147 | 7 | 1.000 | \{x, y, z\}=(x y) z-x(y z) | | |
| 0902.0431_FO0148 | 7 | 1.000 | x, y, z | | |
| 0902.0431_FO0149 | 7 | 0.774 | 10 \quad(a x)(y a)=a(x y) a \quad | | |
| 0902.0431_FO0150 | 7 | 0.957 | 11 R(x y)=R(y x), \quad R(x(y z))=R(y(z x))=R(z(x y))(=R(x y z)) | | |
| 0902.0431_FO0151 | 7 | 0.999 | \left\{1, a_{1}, a_{2}, \cdots, a_{7}\right\} | | |
| 0902.0431_FO0152 | 7 | 0.801 | 12.1 \quad a_{i}\left(a_{j} x\right)=-a_{j}\left(a_{i} x\right), \quad | | |
| 0902.0431_FO0153 | 7 | 0.801 | \quad a_{i} a_{j}=-a_{j} a_{i}, \quad i \neq j | | |
| 0902.0431_FO0154 | 7 | 0.985 | 12.2 \quad a_{i}\left(a_{i} x\right)=-x | | |
| 0902.0431_FO0155 | 7 | 0.985 | \quad a_{i}{ }^{2}=-1 | | |
| 0902.0431_FO0156 | 7 | 1.000 | 12.3 a_{i}\left(a_{j} a_{k}\right)=a_{j}\left(a_{k} a_{i}\right)=a_{k}\left(a_{i} a_{j}\right), \quad i, j, k | | |
| 0902.0431_FO0157 | 7 | 0.999 | \alpha \in G_{2} | | |
| 0902.0431_FO0158 | 7 | 1.000 | x \in \mathfrak{C} | | |
| 0902.0431_FO0159 | 8 | 1.000 | (\alpha 1)(\alpha 1)=\alpha(1 \cdot 1)=\alpha 1 | | |
| 0902.0431_FO0160 | 8 | 1.000 | \alpha 1 \neq 0 | | |
| 0902.0431_FO0161 | 8 | 1.000 | \alpha 1=1 | | |
| 0902.0431_FO0162 | 8 | 1.000 | \left(\alpha e_{i}\right)\left(\alpha e_{i}\right)= | | |
| 0902.0431_FO0163 | 8 | 1.000 | \alpha\left(e_{i} e_{i}\right)=\alpha(-1)=-\alpha 1=-1 | | |
| 0902.0431_FO0164 | 8 | 1.000 | i \neq 0 | | |
| 0902.0431_FO0165 | 8 | 1.000 | \overline{\alpha e_{i}}=-\alpha e_{i} | | |
| 0902.0431_FO0166 | 8 | 1.000 | \alpha \in G_{2}, G_{2} | | |
| 0902.0431_FO0167 | 8 | 1.000 | O(7)=\{\alpha \in O(\mathfrak{C}) \mid \alpha 1=1\} | | |
| 0902.0431_FO0168 | 8 | 1.000 | G_{2} \subset O(7) | | |
| 0902.0431_FO0169 | 8 | 1.000 | \mathfrak{D}_{4} | | |
| 0902.0431_FO0170 | 8 | 1.000 | G_{i j}: \mathfrak{C} \rightarrow \mathfrak{C}, i, j=0,1, \cdots, 7, i \neq j | | |
| 0902.0431_FO0171 | 8 | 1.000 | G_{i j} \in \mathfrak{D}_{4} | | |
| 0902.0431_FO0172 | 8 | 1.000 | \left\{G_{i j} \mid 0 \leq i<j \leq 7\right\} | | |
| 0902.0431_FO0173 | 8 | 1.000 | F_{i j}: \mathfrak{C} \rightarrow \mathfrak{C}, i, j=0,1, \cdots, 7, i \neq j | | |
| 0902.0431_FO0174 | 8 | 0.999 | i, j=0,1, \cdots, 7, i \neq j | | |
| 0902.0431_FO0175 | 8 | 0.999 | F_{i j} \in \mathfrak{D}_{4} | | |
| 0902.0431_FO0176 | 8 | 0.999 | i<j | | |
| 0902.0431_FO0177 | 8 | 1.000 | F_{i j} | | |
| 0902.0431_FO0178 | 8 | 1.000 | G_{i j} | | |
| 0902.0431_FO0179 | 9 | 1.000 | \left\{F_{i j} \mid 0 \leq i<j \leq 7\right\} | | |
| 0902.0431_FO0180 | 9 | 1.000 | \kappa, \pi, \nu: \mathfrak{D}_{4} \rightarrow \mathfrak{D}_{4} | | |
| 0902.0431_FO0181 | 9 | 1.000 | \kappa, \pi, \nu | | |
| 0902.0431_FO0182 | 9 | 1.000 | \kappa^{2}=1, \kappa | | |
| 0902.0431_FO0183 | 9 | 1.000 | \kappa \in \operatorname{Aut}\left(\mathfrak{D}_{4}\right) | | |
| 0902.0431_FO0184 | 9 | 1.000 | \pi | | |
| 0902.0431_FO0185 | 9 | 1.000 | \mathfrak{D}_{4}, \pi | | |
| 0902.0431_FO0186 | 9 | 1.000 | i, j, k, l | | |
| 0902.0431_FO0187 | 9 | 1.000 | i, j, k, l \neq 0 | | |
| 0902.0431_FO0188 | 10 | 1.000 | \pi \in \operatorname{Aut}\left(\mathfrak{D}_{4}\right) | | |
| 0902.0431_FO0189 | 10 | 1.000 | \nu=\pi \kappa | | |
| 0902.0431_FO0190 | 10 | 1.000 | \nu \in \operatorname{Aut}\left(\mathfrak{D}_{4}\right) | | |
| 0902.0431_FO0191 | 10 | 1.000 | a \in \mathfrak{C} | | |
| 0902.0431_FO0192 | 10 | 1.000 | L_{a}, R_{a}, T_{a}: \mathfrak{C} \rightarrow \mathfrak{C} | | |
| 0902.0431_FO0193 | 10 | 1.000 | \mathfrak{C}_{0} | | |
| 0902.0431_FO0194 | 10 | 1.000 | \{a \in \mathfrak{C} \mid \bar{a}=-a\} | | |
| 0902.0431_FO0195 | 10 | 1.000 | a \in \mathfrak{C}_{0} | | |
| 0902.0431_FO0196 | 10 | 1.000 | \left(L_{a} x, y\right)=(a x, y)=(x, \bar{a} y)=-(x, a y)=-\left(x, L_{a} y\right), x, y \in \mathfrak{C} | | |
| 0902.0431_FO0197 | 10 | 1.000 | L_{a} \in \mathfrak{D}_{4} | | |
| 0902.0431_FO0198 | 10 | 1.000 | R_{a} \in \mathfrak{D}_{4} | | |
| 0902.0431_FO0199 | 10 | 1.000 | T_{a}=L_{a}+R_{a} \in \mathfrak{D}_{4} | | |
| 0902.0431_FO0200 | 10 | 1.000 | \left(\kappa L_{a}\right) x=\overline{L_{a} \bar{x}}=\overline{a \bar{x}}=x \bar{a}=-x a=-R_{a} x, x \in \mathfrak{C} | | |
| 0902.0431_FO0201 | 10 | 1.000 | \kappa L_{a}=-R_{a} | | |
| 0902.0431_FO0202 | 10 | 1.000 | a=e_{i}, i=1, \cdots, 7 | | |
| 0902.0431_FO0203 | 11 | 1.000 | \left\{L_{a} \mid a \in \mathfrak{C}_{0}\right\} | | |
| 0902.0431_FO0204 | 11 | 1.000 | D \in \mathfrak{D}_{4} | | |
| 0902.0431_FO0205 | 11 | 1.000 | \mathfrak{D}^{\prime} | | |
| 0902.0431_FO0206 | 11 | 0.996 | L_{e_{i}}=2 F_{i 0} | | |
| 0902.0431_FO0207 | 11 | 0.996 | \left[L_{e_{i}}, L_{e_{j}}\right]=4\left[F_{i 0}, F_{j 0}\right]=-4 F_{i j}, i \neq 0, j \neq 0, i \neq j | | |
| 0902.0431_FO0208 | 11 | 1.000 | F_{i j}, i, j=0,1, \cdots, 7, i \neq j | | |
| 0902.0431_FO0209 | 11 | 1.000 | \left\{F_{i j}, i<j\right\} | | |
| 0902.0431_FO0210 | 11 | 1.000 | \operatorname{Aut}\left(\mathfrak{D}_{4}\right) | | |
| 0902.0431_FO0211 | 11 | 1.000 | \mathfrak{S}_{3} | | |
| 0902.0431_FO0212 | 11 | 0.989 | \kappa | | |
| 0902.0431_FO0213 | 11 | 0.989 | S_{3} | | |
| 0902.0431_FO0214 | 11 | 1.000 | \kappa^{2}=1, \pi^{2}=1, \nu^{3}=1 | | |
| 0902.0431_FO0215 | 11 | 1.000 | L_{a} | | |
| 0902.0431_FO0216 | 11 | 1.000 | f: \mathfrak{S}_{3} \rightarrow S_{3} | | |
| 0902.0431_FO0217 | 11 | 0.999 | D_{1} \in | | |
| 0902.0431_FO0218 | 11 | 1.000 | D_{2}, D_{3} \in \mathfrak{D}_{4} | | |
| 0902.0431_FO0219 | 11 | 1.000 | D_{2}, D_{3} | | |
| 0902.0431_FO0220 | 11 | 1.000 | D_{1} | | |
| 0902.0431_FO0221 | 11 | 1.000 | L_{a}, R_{a}, T_{a} \in \mathfrak{D}_{4} | | |
| 0902.0431_FO0222 | 11 | 1.000 | (a x) y+x(y a)=a(x y)+(x y) a | | |
| 0902.0431_FO0223 | 12 | 1.000 | b \in \mathfrak{C}_{0} | | |
| 0902.0431_FO0224 | 12 | 1.000 | T_{a} | | |
| 0902.0431_FO0225 | 12 | 1.000 | a | | |
| 0902.0431_FO0226 | 12 | 1.000 | b | | |
| 0902.0431_FO0227 | 12 | 1.000 | D_{1} \in \mathfrak{D}_{4} | | |
| 0902.0431_FO0228 | 12 | 1.000 | D_{2}=R_{a}+\sum\left[R_{b}, R_{c}\right], D_{3}=T_{a}+\sum\left[T_{b}, T_{c}\right] | | |
| 0902.0431_FO0229 | 12 | 1.000 | D_{2} | | |
| 0902.0431_FO0230 | 12 | 1.000 | D_{3} | | |
| 0902.0431_FO0231 | 12 | 1.000 | D_{1}=0 | | |
| 0902.0431_FO0232 | 12 | 1.000 | D_{2}=D_{3}=0 | | |
| 0902.0431_FO0233 | 12 | 1.000 | x=1 | | |
| 0902.0431_FO0234 | 12 | 1.000 | D_{2} y=D_{3} y | | |
| 0902.0431_FO0235 | 12 | 1.000 | D_{2}=D_{3}(=D) | | |
| 0902.0431_FO0236 | 12 | 1.000 | D 1=p | | |
| 0902.0431_FO0237 | 12 | 1.000 | 2(p, 1)=(p, 1)+(1, p)=(D 1,1)+(1, D 1)=0 | | |
| 0902.0431_FO0238 | 12 | 0.997 | p \in \mathfrak{C}_{0} | | |
| 0902.0431_FO0239 | 12 | 0.997 | y=1 | | |
| 0902.0431_FO0240 | 12 | 0.997 | x p=D x | | |
| 0902.0431_FO0241 | 12 | 0.997 | p \in \boldsymbol{R} | | |
| 0902.0431_FO0242 | 12 | 0.997 | p=0 | | |
| 0902.0431_FO0243 | 12 | 0.997 | D x=x p=0 | | |
| 0902.0431_FO0244 | 12 | 0.999 | D=0 | | |
| 0902.0431_FO0245 | 12 | 1.000 | D_{1}=L_{a}+\sum\left[L_{b}, L_{c}\right], a, b, c \in \mathfrak{C}_{0} | | |
| 0902.0431_FO0246 | 12 | 1.000 | D_{2}=\nu D_{1}, D_{3}=\pi D_{1} | | |
| 0902.0431_FO0247 | 12 | 1.000 | D_{1}, D_{2}, D_{3} \in \mathfrak{D}_{4} | | |
| 0902.0431_FO0248 | 13 | 1.000 | D_{2} \in \mathfrak{D}_{4} | | |
| 0902.0431_FO0249 | 13 | 1.000 | D_{3}{ }^{\prime}, D_{1}{ }^{\prime} \in \mathfrak{D}_{4} | | |
| 0902.0431_FO0250 | 13 | 1.000 | D_{3}{ }^{\prime}=\nu D_{2}, \kappa D_{1}{ }^{\prime}=\pi D_{2} | | |
| 0902.0431_FO0251 | 13 | 1.000 | D_{2}=\nu D_{1}, \kappa D_{3}=\pi D_{1} | | |
| 0902.0431_FO0252 | 13 | 1.000 | D \in \operatorname{Hom}_{\boldsymbol{R}}(\mathfrak{C}) | | |
| 0902.0431_FO0253 | 13 | 1.000 | (\exp t D)(x y)=((\exp t D) x)((\exp t D) y), t \in \boldsymbol{R} | | |
| 0902.0431_FO0254 | 13 | 1.000 | t | | |
| 0902.0431_FO0255 | 13 | 1.000 | t=0 | | |
| 0902.0431_FO0256 | 13 | 1.000 | D(x y)=(D x) y+ | | |
| 0902.0431_FO0257 | 13 | 1.000 | x(D y) | | |
| 0902.0431_FO0258 | 13 | 1.000 | D(x y)=(D x) y+x(D y) | | |
| 0902.0431_FO0259 | 13 | 1.000 | \alpha=\exp t D | | |
| 0902.0431_FO0260 | 13 | 1.000 | \alpha(x y)=(\alpha x)(\alpha y) | | |
| 0902.0431_FO0261 | 13 | 0.997 | \mathfrak{b}_{3}=\mathfrak{s o}(7) | | |
| 0902.0431_FO0262 | 13 | 0.987 | S O(7) | | |
| 0902.0431_FO0263 | 13 | 1.000 | \mathfrak{b}_{3} | | |
| 0902.0431_FO0264 | 13 | 1.000 | D \in \mathfrak{g}_{2} | | |
| 0902.0431_FO0265 | 13 | 1.000 | x=y=1 | | |
| 0902.0431_FO0266 | 13 | 1.000 | D 1=0 | | |
| 0902.0431_FO0267 | 13 | 0.997 | \left(D e_{i}\right) e_{i}+e_{i}\left(D e_{i}\right)=D\left(e_{i} e_{i}\right)=D(-1)=0 | | |
| 0902.0431_FO0268 | 13 | 0.997 | D e_{i} \in \mathfrak{C}_{0} | | |
| 0902.0431_FO0269 | 13 | 0.999 | D x \in \mathfrak{C}_{0}, x \in \mathfrak{C} | | |
| 0902.0431_FO0270 | 13 | 0.998 | D | | |
| 0902.0431_FO0271 | 13 | 0.998 | (D x) y+x(D y)+(D y) x+y(D x)=0 | | |
| 0902.0431_FO0272 | 13 | 1.000 | D x, D y \in \mathfrak{C}_{0} | | |
| 0902.0431_FO0273 | 14 | 1.000 | \mathfrak{g}_{2} \subset \mathfrak{b}_{3} | | |
| 0902.0431_FO0274 | 14 | 0.998 | D \in \mathfrak{b}_{3} | | |
| 0902.0431_FO0275 | 14 | 0.998 | D=\sum_{0<i<j} \lambda_{i j} G_{i j}, \lambda_{i j} \in \boldsymbol{R} | | |
| 0902.0431_FO0276 | 14 | 0.996 | \pi D=D | | |
| 0902.0431_FO0277 | 14 | 1.000 | 1 \in \mathfrak{C} | | |
| 0902.0431_FO0278 | 14 | 1.000 | e_{j} e_{i} | | |
| 0902.0431_FO0279 | 14 | 1.000 | e_{k} | | |
| 0902.0431_FO0280 | 14 | 1.000 | e_{1}, e_{2}, \cdots, e_{7} | | |
| 0902.0431_FO0281 | 14 | 1.000 | \boldsymbol{C} | | |
| 0902.0431_FO0282 | 15 | 1.000 | \boldsymbol{C} \oplus \boldsymbol{C}^{3} | | |
| 0902.0431_FO0283 | 15 | 1.000 | (\boldsymbol{m}, \boldsymbol{n}) | | |
| 0902.0431_FO0284 | 15 | 1.000 | \langle\boldsymbol{m}, \boldsymbol{n}\rangle | | |
| 0902.0431_FO0285 | 15 | 1.000 | \boldsymbol{m} \times \boldsymbol{n} | | |
| 0902.0431_FO0286 | 15 | 0.786 | \boldsymbol{m}=\left(\begin{array}{c}m_{1} \\ m_{2} \\ m_{3}\end{array}\right), \boldsymbol{n}=\left(\begin{array}{c}n_{1} \\ n_{2} \\ n_{3}\end{array}\right) \in \boldsymbol{C}^{3} | | |
| 0902.0431_FO0287 | 15 | 1.000 | \left(\mathfrak{g}_{2}\right)_{e_{1}} | | |
| 0902.0431_FO0288 | 15 | 0.951 | \quad\left(\mathfrak{g}_{2}\right)_{e_{1}} \cong \mathfrak{s u}(3) | | |
| 0902.0431_FO0289 | 15 | 1.000 | \varphi_{*}: \mathfrak{s u}(3)=\left\{D \in M(3, \boldsymbol{C}) \mid D^{*}=-D, \operatorname{tr}(D)=\right. | | |
| 0902.0431_FO0290 | 15 | 1.000 | 0\} \rightarrow\left(\mathfrak{g}_{2}\right)_{e_{1}} | | |
| 0902.0431_FO0291 | 15 | 1.000 | \varphi_{*}(D) \in\left(\mathfrak{g}_{2}\right)_{e_{1}} | | |
| 0902.0431_FO0292 | 16 | 0.671 | \mathfrak{s} \mathfrak{u}(3) | | |
| 0902.0431_FO0293 | 16 | 0.671 | E_{k l} \in M(3, \boldsymbol{R}) | | |
| 0902.0431_FO0294 | 16 | 0.671 | (k, l) | | |
| 0902.0431_FO0295 | 16 | 1.000 | \varphi_{*}(D) \subset \mathfrak{g}_{2} | | |
| 0902.0431_FO0296 | 16 | 1.000 | \varphi_{*}(D) e_{1}=0 | | |
| 0902.0431_FO0297 | 16 | 1.000 | \varphi_{*}(D) \subset\left(\mathfrak{g}_{2}\right)_{e_{1}} | | |
| 0902.0431_FO0298 | 16 | 0.998 | \varphi_{*}: \mathfrak{s u}(3) \rightarrow \mathfrak{g}_{2} | | |
| 0902.0431_FO0299 | 16 | 1.000 | \varphi_{*}(\mathfrak{s u}(3) | | |
| 0902.0431_FO0300 | 16 | 1.000 | \mathfrak{S} | | |
| 0902.0431_FO0301 | 16 | 1.000 | S_{1}, \cdots, S_{6} | | |
| 0902.0431_FO0302 | 16 | 0.999 | \varphi_{*}: \mathfrak{s u}(3) \rightarrow\left(\mathfrak{g}_{2}\right)_{e_{1}} | | |
| 0902.0431_FO0303 | 16 | 0.999 | B \in\left(\mathfrak{g}_{2}\right)_{e_{1}} | | |
| 0902.0431_FO0304 | 16 | 1.000 | B e_{1}=0 | | |
| 0902.0431_FO0305 | 16 | 1.000 | x_{1}=\cdots=x_{6}=0 | | |
| 0902.0431_FO0306 | 16 | 1.000 | B=D \in \mathfrak{s u}(3) | | |
| 0902.0431_FO0307 | 16 | 1.000 | \mathfrak{C}^{C}=\left\{x_{1}+i x_{2} \mid x_{1}, x_{2} \in \mathfrak{C}\right\} | | |
| 0902.0431_FO0308 | 16 | 1.000 | \mathfrak{C}^{C} | | |
| 0902.0431_FO0309 | 16 | 1.000 | x y | | |
| 0902.0431_FO0310 | 16 | 0.535 | 1 \sim 12.3 | | |
| 0902.0431_FO0311 | 17 | 1.000 | \mathfrak{s u}(3) \subset \mathfrak{g}_{2} | | |
| 0902.0431_FO0312 | 17 | 1.000 | [D, S], D \in \mathfrak{s u}(3), S \in \mathfrak{S} | | |
| 0902.0431_FO0318 | 17 | 1.000 | H_{1} | | |
| 0902.0431_FO0326 | 17 | 1.000 | L_{13} | | |
| 0902.0431_FO0328 | 17 | 1.000 | L_{31} | | |
| 0902.0431_FO0331 | 17 | 0.964 | W | | |
| 0902.0431_FO0332 | 17 | 1.000 | S_{k} | | |
| 0902.0431_FO0333 | 17 | 1.000 | S_{k}, k=1,2, \cdots, 6 | | |
| 0902.0431_FO0334 | 17 | 1.000 | W=\mathfrak{S} | | |
| 0902.0431_FO0335 | 17 | 1.000 | S=\sum_{k=1}^{6} x_{k} S_{k}, x_{k} \in \boldsymbol{R} | | |
| 0902.0431_FO0336 | 17 | 1.000 | x_{1} \neq 0 | | |
| 0902.0431_FO0337 | 17 | 1.000 | S | | |
| 0902.0431_FO0338 | 17 | 1.000 | x_{1} S_{2}-x_{2} S_{1}-x_{3} S_{4}+x_{4} S_{3} \in W | | |
| 0902.0431_FO0340 | 146 | 0.909 | ) | | |
| 0902.0431_FO0341 | 17 | 0.877 | \times x_{2} | | |
| 0902.0431_FO0342 | 17 | 0.877 | \times x_{1} | | |
| 0902.0431_FO0343 | 17 | 0.877 | \left(x_{1}{ }^{2}+x_{2}{ }^{2}\right) S_{5} \in W | | |
| 0902.0431_FO0344 | 17 | 0.877 | x_{1}{ }^{2}+x_{2}{ }^{2} \neq 0 | | |
| 0902.0431_FO0345 | 17 | 1.000 | S_{5} \in W | | |
| 0902.0431_FO0346 | 17 | 0.986 | [\mathfrak{s u}(3), \mathfrak{S}] | | |
| 0902.0431_FO0347 | 17 | 0.967 | [\mathfrak{s u}(3), \mathfrak{S}]=\mathfrak{S} | | |
| 0902.0431_FO0348 | 17 | 0.995 | p: \mathfrak{g}_{2} \rightarrow \mathfrak{s u}(3) | | |
| 0902.0431_FO0349 | 17 | 0.995 | q: \mathfrak{g}_{2} \rightarrow \mathfrak{S} | | |
| 0902.0431_FO0350 | 17 | 0.995 | \mathfrak{g}_{2}=\mathfrak{s u}(3) \oplus \mathfrak{S} | | |
| 0902.0431_FO0351 | 17 | 0.995 | \mathfrak{a} | | |
| 0902.0431_FO0352 | 17 | 1.000 | p(\mathfrak{a}) | | |
| 0902.0431_FO0353 | 17 | 1.000 | D \in p(\mathfrak{a}) | | |
| 0902.0431_FO0354 | 17 | 0.998 | S \in \mathfrak{S} | | |
| 0902.0431_FO0355 | 17 | 0.998 | D+S \in \mathfrak{a} | | |
| 0902.0431_FO0356 | 17 | 0.998 | D^{\prime} \in \mathfrak{s u}(3) | | |
| 0902.0431_FO0357 | 18 | 1.000 | \left[D^{\prime}, D\right] \in p(\mathfrak{a}) | | |
| 0902.0431_FO0358 | 18 | 0.686 | \mathfrak{s u}(3) \cap \mathfrak{a} \neq\{0\} | | |
| 0902.0431_FO0359 | 18 | 0.686 | \mathfrak{S} \cap \mathfrak{a} \neq\{0\} | | |
| 0902.0431_FO0360 | 18 | 0.686 | \mathfrak{s} \mathfrak{u}(3) \cap \mathfrak{a}=\{0\} | | |
| 0902.0431_FO0361 | 18 | 0.928 | \mathfrak{S} \cap \mathfrak{a}=\{0\} | | |
| 0902.0431_FO0362 | 18 | 0.928 | p \mid \mathfrak{a}: \mathfrak{a} \rightarrow \mathfrak{s} \mathfrak{u}(3) | | |
| 0902.0431_FO0363 | 18 | 0.985 | p(\mathfrak{a})=\mathfrak{s} \mathfrak{u}(3) | | |
| 0902.0431_FO0364 | 18 | 0.994 | \operatorname{dim} \mathfrak{a}=\operatorname{dim} p(\mathfrak{a})=\operatorname{dim} \mathfrak{s} \mathfrak{u}(3)=8 | | |
| 0902.0431_FO0365 | 18 | 1.000 | q \mid \mathfrak{a}: \mathfrak{a} \rightarrow \mathfrak{S} | | |
| 0902.0431_FO0366 | 18 | 1.000 | \operatorname{dim} \mathfrak{a} \leq \operatorname{dim} \mathfrak{S}=6 | | |
| 0902.0431_FO0367 | 18 | 0.919 | \mathfrak{s} \mathfrak{u}(3) \cap \mathfrak{a}=\mathfrak{s} \mathfrak{u}(3) | | |
| 0902.0431_FO0368 | 18 | 0.984 | \mathfrak{a} \supset \mathfrak{s u}(3) | | |
| 0902.0431_FO0369 | 18 | 0.977 | \mathfrak{a} \supset \mathfrak{s u}(3) \oplus \mathfrak{S}=\mathfrak{g}_{2} | | |
| 0902.0431_FO0370 | 18 | 1.000 | S \in \mathfrak{S} \cap \mathfrak{a} \subset \mathfrak{a} | | |
| 0902.0431_FO0371 | 18 | 0.910 | S_{1} \in \mathfrak{a} | | |
| 0902.0431_FO0372 | 18 | 0.910 | 0 \neq 4 H_{1}+2 H_{2}=\left[S_{1}, S_{2}\right] \in \mathfrak{a} | | |
| 0902.0431_FO0373 | 18 | 1.000 | \mathfrak{a}=\mathfrak{g}_{2} | | |
| 0902.0431_FO0374 | 18 | 0.895 | L_{i j}, L_{j i} | | |
| 0902.0431_FO0375 | 18 | 1.000 | B_{2} | | |
| 0902.0431_FO0376 | 18 | 1.000 | \operatorname{tr}\left(D_{1} D_{2}\right) | | |
| 0902.0431_FO0377 | 18 | 0.999 | k \in C | | |
| 0902.0431_FO0378 | 18 | 1.000 | k | | |
| 0902.0431_FO0379 | 18 | 1.000 | D_{1}=D_{2}=H_{1} | | |
| 0902.0431_FO0380 | 19 | 1.000 | \operatorname{tr}\left(H_{1} H_{1}\right)=(-1) \times 4=-4 | | |
| 0902.0431_FO0381 | 19 | 1.000 | k=4 | | |
| 0902.0431_FO0382 | 19 | 0.984 | f_{*}: \mathfrak{s l}(3, C) \rightarrow \mathfrak{s u}(3)^{C} | | |
| 0902.0431_FO0383 | 19 | 0.984 | \varphi_{*} | | |
| 0902.0431_FO0384 | 19 | 1.000 | \mathfrak{s} \mathfrak{u}(3)^{C} \rightarrow \mathfrak{g}_{2}{ }^{C} | | |
| 0902.0431_FO0385 | 19 | 1.000 | \mathfrak{s l}(3, C) | | |
| 0902.0431_FO0386 | 19 | 1.000 | f_{*} | | |
| 0902.0431_FO0387 | 19 | 1.000 | \pm\left(\lambda_{k}-\lambda_{l}\right), 1 \leq k<l \leq 3 | | |
| 0902.0431_FO0388 | 19 | 1.000 | \mathfrak{s l}(3, \boldsymbol{C}) | | |
| 0902.0431_FO0389 | 19 | 1.000 | E_{k l} | | |
| 0902.0431_FO0390 | 19 | 1.000 | \lambda_{k}-\lambda_{l} | | |
| 0902.0431_FO0391 | 19 | 1.000 | \lambda_{1}+\lambda_{2}+\lambda_{3}=0 | | |
| 0902.0431_FO0392 | 19 | 0.999 | \quad \mathfrak{h}=\left\{-i \lambda_{1} G_{23}-i \lambda_{2} G_{45}-i \lambda_{3} G_{67} \in \mathfrak{g}_{2}{ }^{C} \mid \lambda_{k} \in C\right\} \subset \mathfrak{s l}(3, C) \subset \mathfrak{g}_{2}{ }^{C} | | |
| 0902.0431_FO0393 | 19 | 0.777 | \mathfrak{s} \mathfrak{l}(3, C) | | |
| 0902.0431_FO0394 | 20 | 1.000 | \Pi=\left\{\alpha_{1}, \alpha_{2}\right\} | | |
| 0902.0431_FO0395 | 20 | 1.000 | \mathfrak{h}_{\boldsymbol{R}} | | |
| 0902.0431_FO0396 | 20 | 1.000 | \mathfrak{h} | | |
| 0902.0431_FO0397 | 20 | 1.000 | H=-i \lambda_{1} G_{23}-i \lambda_{2} G_{45}-i \lambda_{3} G_{67}, H^{\prime}=-i \lambda_{1}{ }^{\prime} G_{23}-i \lambda_{2}{ }^{\prime} G_{45}-i \lambda_{3}{ }^{\prime} G_{67} \in \mathfrak{h}_{\boldsymbol{R}} | | |
| 0902.0431_FO0398 | 20 | 0.996 | H_{\alpha_{i}} \in \mathfrak{h}_{\boldsymbol{R}} | | |
| 0902.0431_FO0399 | 20 | 0.996 | \alpha_{i}\left(B_{2}\left(H_{\alpha}, H\right)=\right. | | |
| 0902.0431_FO0400 | 20 | 0.531 | \alpha(H), H \in \mathfrak{h}_{\boldsymbol{R}} | | |
| 0902.0431_FO0401 | 21 | 1.000 | C_{1} \oplus C_{1} | | |
| 0902.0431_FO0402 | 21 | 1.000 | A_{2} | | |
| 0902.0431_FO0403 | 21 | 1.000 | \left(G_{2}\right)_{e_{1}} | | |
| 0902.0431_FO0404 | 21 | 0.946 | \quad\left(G_{2}\right)_{e_{1}} \cong S U(3) | | |
| 0902.0431_FO0405 | 21 | 0.998 | \varphi: S U(3) \rightarrow\left(G_{2}\right)_{e_{1}} | | |
| 0902.0431_FO0406 | 21 | 1.000 | \varphi(A) \in\left(G_{2}\right)_{e_{1}} | | |
| 0902.0431_FO0407 | 21 | 1.000 | \alpha=\varphi(A), A \in S U(3) | | |
| 0902.0431_FO0408 | 21 | 1.000 | x=a+\boldsymbol{m}, y= | | |
| 0902.0431_FO0409 | 21 | 0.850 | b+\boldsymbol{n} \in \boldsymbol{C} \oplus \boldsymbol{C}^{3}=\mathfrak{C} | | |
| 0902.0431_FO0410 | 21 | 0.850 | A \in S U(3) | | |
| 0902.0431_FO0411 | 21 | 0.850 | \widetilde{A} | | |
| 0902.0431_FO0412 | 21 | 1.000 | A)=A^{-1}=A^{*} | | |
| 0902.0431_FO0413 | 21 | 1.000 | \varphi(A) \in G_{2} | | |
| 0902.0431_FO0414 | 21 | 1.000 | \varphi(A) e_{1}=e_{1} | | |
| 0902.0431_FO0415 | 21 | 1.000 | \varphi | | |
| 0902.0431_FO0416 | 21 | 1.000 | \alpha \in\left(G_{2}\right)_{e_{1}} | | |
| 0902.0431_FO0417 | 21 | 1.000 | \alpha | | |
| 0902.0431_FO0418 | 21 | 1.000 | \boldsymbol{C}^{3} | | |
| 0902.0431_FO0419 | 21 | 0.996 | A=\left(\boldsymbol{a}_{1}, \boldsymbol{a}_{2}, \boldsymbol{a}_{3}\right) \in M(3, \boldsymbol{C}) | | |
| 0902.0431_FO0420 | 21 | 0.996 | \left(\alpha e_{2}\right)\left(\alpha e_{4}\right)=\alpha\left(e_{2} e_{4}\right)= | | |
| 0902.0431_FO0421 | 21 | 1.000 | -\alpha e_{6} | | |
| 0902.0431_FO0422 | 21 | 1.000 | \boldsymbol{a}_{1} \boldsymbol{a}_{2}=-\boldsymbol{a}_{3} | | |
| 0902.0431_FO0423 | 21 | 1.000 | -\left\langle\boldsymbol{a}_{1}, \boldsymbol{a}_{2}\right\rangle-\overline{\boldsymbol{a}_{1} \times \boldsymbol{a}_{2}}=-\boldsymbol{a}_{3} | | |
| 0902.0431_FO0424 | 22 | 1.000 | \left\langle\boldsymbol{a}_{2}, \boldsymbol{a}_{3}\right\rangle=\left\langle\boldsymbol{a}_{3}, \boldsymbol{a}_{1}\right\rangle=0 | | |
| 0902.0431_FO0425 | 22 | 1.000 | \left(\alpha e_{k}\right)\left(\alpha e_{k}\right)=\alpha\left(e_{k} e_{k}\right)= | | |
| 0902.0431_FO0426 | 22 | 1.000 | \alpha(-1)=-1 | | |
| 0902.0431_FO0427 | 22 | 1.000 | \left\langle\boldsymbol{a}_{k}, \boldsymbol{a}_{k}\right\rangle=1 | | |
| 0902.0431_FO0428 | 22 | 1.000 | A \in U(3) | | |
| 0902.0431_FO0429 | 22 | 1.000 | \operatorname{det} A=\left(\boldsymbol{a}_{3}, \boldsymbol{a}_{1} \times\right. | | |
| 0902.0431_FO0430 | 22 | 0.776 | \left.\boldsymbol{a}_{2}\right)=\left(\boldsymbol{a}_{3}, \overline{\boldsymbol{a}}_{3}\right)=\left\langle\boldsymbol{a}_{3}, \boldsymbol{a}_{3}\right\rangle=1 | | |
| 0902.0431_FO0431 | 22 | 0.776 | (\boldsymbol{a}, \boldsymbol{b}) | | |
| 0902.0431_FO0432 | 22 | 0.888 | \left.(\boldsymbol{a}, \boldsymbol{b})={ }^{t} \boldsymbol{a} \boldsymbol{b}\right) | | |
| 0902.0431_FO0433 | 22 | 0.888 | \varphi(A)=\alpha | | |
| 0902.0431_FO0434 | 22 | 0.984 | \operatorname{Ker} \varphi=\{E\} | | |
| 0902.0431_FO0435 | 22 | 0.984 | \operatorname{SU}(3) \cong\left(G_{2}\right)_{e_{1}} | | |
| 0902.0431_FO0436 | 22 | 0.961 | G_{2} / S U(3) \simeq S^{6} | | |
| 0902.0431_FO0437 | 22 | 1.000 | S^{6}=\{a \in \mathfrak{C}|\bar{a}=-a,|a|=1\} | | |
| 0902.0431_FO0438 | 22 | 1.000 | S^{6} | | |
| 0902.0431_FO0439 | 22 | 1.000 | a \in S^{6} | | |
| 0902.0431_FO0440 | 22 | 1.000 | e_{1} \in S^{6} | | |
| 0902.0431_FO0441 | 22 | 1.000 | a_{1} \in S^{6} | | |
| 0902.0431_FO0442 | 22 | 1.000 | a_{2} \in S^{6} | | |
| 0902.0431_FO0443 | 22 | 1.000 | \left(a_{1}, a_{2}\right)=0 | | |
| 0902.0431_FO0444 | 22 | 1.000 | a_{3} \in S^{6} | | |
| 0902.0431_FO0445 | 22 | 1.000 | a_{3} | | |
| 0902.0431_FO0446 | 22 | 1.000 | \left(a_{1}, a_{3}\right)=\left(a_{2}, a_{3}\right)=0 | | |
| 0902.0431_FO0447 | 22 | 1.000 | a_{4} \in S^{6} | | |
| 0902.0431_FO0448 | 22 | 1.000 | \left(a_{1}, a_{4}\right)=\left(a_{2}, a_{4}\right)=\left(a_{3}, a_{4}\right)=0 | | |
| 0902.0431_FO0449 | 22 | 0.996 | \left\{a_{0}=1, a_{1}, a_{2}, \cdots, a_{7}\right\} | | |
| 0902.0431_FO0450 | 22 | 0.996 | \left|a_{i}\right|= | | |
| 0902.0431_FO0451 | 22 | 1.000 | 1,0 \leq i \leq 7 | | |
| 0902.0431_FO0452 | 22 | 1.000 | \left(a_{i}, a_{j}\right)=0, i \neq j | | |
| 0902.0431_FO0453 | 22 | 0.998 | \left\{e_{0}=1, e_{1}, e_{2}, \cdots, e_{7}\right\} | | |
| 0902.0431_FO0454 | 22 | 1.000 | \alpha: \mathfrak{C} \rightarrow \mathfrak{C} | | |
| 0902.0431_FO0455 | 22 | 0.844 | O(7) | | |
| 0902.0431_FO0456 | 22 | 0.844 | \alpha \in O(7) | | |
| 0902.0431_FO0457 | 23 | 1.000 | \alpha e_{1}=a_{1} | | |
| 0902.0431_FO0458 | 23 | 1.000 | \alpha^{-1} a_{1}=e_{1} | | |
| 0902.0431_FO0459 | 23 | 0.942 | e_{1} | | |
| 0902.0431_FO0460 | 23 | 1.000 | G_{2}=\left\{\alpha \in \operatorname{Iso}_{\boldsymbol{R}}(\mathfrak{C}) \mid \alpha(x y)=(\alpha x)(\alpha y)\right\} | | |
| 0902.0431_FO0461 | 23 | 1.000 | S O(7)=\{\alpha \in O(7) \mid \operatorname{det} \alpha= | | |
| 0902.0431_FO0462 | 23 | 0.975 | 1\}: G_{2} \subset S O(7) | | |
| 0902.0431_FO0463 | 23 | 1.000 | \operatorname{dim} G_{2}= | | |
| 0902.0431_FO0464 | 23 | 0.999 | \operatorname{dim} \mathfrak{g}_{2}=14 | | |
| 0902.0431_FO0465 | 23 | 0.761 | \operatorname{SU}(3) | | |
| 0902.0431_FO0466 | 23 | 1.000 | \operatorname{dim}\left(G_{2} /\left(G_{2}\right)_{e_{1}}\right)=\operatorname{dim} G_{2}-\operatorname{dim} S U(3)=14-8=6=\operatorname{dim} S^{6} | | |
| 0902.0431_FO0467 | 23 | 0.997 | \varphi: S U(3) \rightarrow G_{2} | | |
| 0902.0431_FO0468 | 23 | 0.997 | w: \mathfrak{C} \rightarrow \mathfrak{C} | | |
| 0902.0431_FO0469 | 23 | 0.996 | \omega_{1}=-\frac{1}{2}+\frac{\sqrt{3}}{2} e_{1} \in \boldsymbol{C} \subset \mathfrak{C} | | |
| 0902.0431_FO0470 | 23 | 1.000 | w \in G_{2} | | |
| 0902.0431_FO0471 | 23 | 1.000 | w^{3}=1 | | |
| 0902.0431_FO0472 | 23 | 1.000 | \left(G_{2}\right)^{w} | | |
| 0902.0431_FO0473 | 23 | 0.996 | \quad\left(G_{2}\right)^{w}=\left(G_{2}\right)_{e_{1}} \cong S U(3) | | |
| 0902.0431_FO0474 | 23 | 0.997 | \varphi(S U(3)) \in\left(G_{2}\right)^{w} | | |
| 0902.0431_FO0475 | 23 | 0.997 | a+\boldsymbol{m} \in \boldsymbol{C} \oplus \boldsymbol{C}^{3}=\mathfrak{C} | | |
| 0902.0431_FO0476 | 23 | 1.000 | w \varphi(A)=\varphi(A) w | | |
| 0902.0431_FO0477 | 23 | 1.000 | \varphi(A) \in\left(G_{2}\right)^{w} | | |
| 0902.0431_FO0478 | 23 | 1.000 | \alpha \in\left(G_{2}\right)^{w} | | |
| 0902.0431_FO0479 | 23 | 1.000 | \mathfrak{C}_{w}=\{x \in \mathfrak{C} \mid w x=x\} | | |
| 0902.0431_FO0480 | 23 | 1.000 | \mathfrak{C}_{w}=\boldsymbol{C} | | |
| 0902.0431_FO0481 | 23 | 1.000 | w \alpha=\alpha w, \mathfrak{C}_{w} | | |
| 0902.0431_FO0482 | 23 | 1.000 | \mathfrak{C}_{w} | | |
| 0902.0431_FO0483 | 24 | 0.999 | \gamma_{1}: \mathfrak{C} \rightarrow \mathfrak{C} | | |
| 0902.0431_FO0484 | 24 | 0.999 | \gamma_{1}(a+\boldsymbol{m})=\bar{a}+\overline{\boldsymbol{m}} | | |
| 0902.0431_FO0485 | 24 | 1.000 | \gamma_{1} \in G_{2} | | |
| 0902.0431_FO0486 | 24 | 1.000 | \gamma_{1} e_{1}=-e_{1} | | |
| 0902.0431_FO0487 | 24 | 1.000 | \beta=\gamma_{1} \alpha | | |
| 0902.0431_FO0488 | 24 | 1.000 | \beta e_{1}=e_{1} | | |
| 0902.0431_FO0489 | 24 | 0.940 | \beta \in S U(3) \subset\left(G_{2}\right)_{e_{1}} | | |
| 0902.0431_FO0490 | 24 | 0.940 | \subset\left(G_{2}\right)^{w} | | |
| 0902.0431_FO0491 | 24 | 0.940 | \gamma_{1}=\beta \alpha^{-1} \in\left(G_{2}\right)^{w} | | |
| 0902.0431_FO0492 | 24 | 0.966 | \alpha e_{1}=e_{1} | | |
| 0902.0431_FO0493 | 24 | 0.966 | \alpha \in S U(3) | | |
| 0902.0431_FO0494 | 24 | 0.905 | \left(G_{2}\right)^{w}=\left(G_{2}\right)_{e_{1}} | | |
| 0902.0431_FO0495 | 24 | 1.000 | \boldsymbol{H} | | |
| 0902.0431_FO0496 | 24 | 0.996 | \boldsymbol{H} \oplus \boldsymbol{H} e_{4} | | |
| 0902.0431_FO0497 | 24 | 1.000 | \left(G_{2}\right)^{\gamma} | | |
| 0902.0431_FO0498 | 24 | 0.551 | \left(G_{2}\right)^{\gamma} \cong(S p(1) \times S p(1)) / \boldsymbol{Z}_{2}, \boldsymbol{Z}_{2}=\{(1,1),(-1,-1)\} | | |
| 0902.0431_FO0499 | 24 | 0.999 | \varphi: \operatorname{Sp}(1) \times \operatorname{Sp}(1) \rightarrow\left(G_{2}\right)^{\gamma} | | |
| 0902.0431_FO0500 | 25 | 0.973 | \varphi(p, q) \in\left(G_{2}\right)^{\gamma} | | |
| 0902.0431_FO0501 | 25 | 0.973 | \alpha=\varphi(p, q), p, q \in S p(1) | | |
| 0902.0431_FO0502 | 25 | 0.973 | x=m+a e_{4} | | |
| 0902.0431_FO0503 | 25 | 0.999 | y=n+b e_{4} \in \boldsymbol{H} \oplus \boldsymbol{H} e_{4}=\mathfrak{C} | | |
| 0902.0431_FO0504 | 25 | 1.000 | \varphi(p, q) \in G_{2} | | |
| 0902.0431_FO0505 | 25 | 1.000 | \gamma \varphi(p, q)=\varphi(p, q) \gamma | | |
| 0902.0431_FO0506 | 25 | 1.000 | \alpha \in\left(G_{2}\right)^{\gamma} | | |
| 0902.0431_FO0507 | 25 | 1.000 | \gamma \alpha=\alpha \gamma, \mathfrak{C}_{\gamma}=\{x \in \mathfrak{C} \mid \gamma x=x\}=\boldsymbol{H} | | |
| 0902.0431_FO0508 | 25 | 1.000 | \mathfrak{C}_{\gamma} | | |
| 0902.0431_FO0509 | 25 | 1.000 | q \in \operatorname{Sp}(1) | | |
| 0902.0431_FO0510 | 25 | 1.000 | \beta=\varphi(1, q)^{-1} \alpha | | |
| 0902.0431_FO0511 | 25 | 1.000 | \beta \in\left(G_{2}\right)^{\gamma} | | |
| 0902.0431_FO0512 | 25 | 0.994 | \beta \mid \boldsymbol{H}=1 | | |
| 0902.0431_FO0513 | 25 | 0.994 | \beta | | |
| 0902.0431_FO0514 | 25 | 0.994 | \mathfrak{C}_{-\gamma}=\{x \in \mathfrak{C} \mid \gamma x=-x\}=\boldsymbol{H} e_{4} | | |
| 0902.0431_FO0515 | 25 | 1.000 | \beta e_{4}=p e_{4} | | |
| 0902.0431_FO0516 | 25 | 1.000 | p \in \boldsymbol{H} | | |
| 0902.0431_FO0517 | 25 | 1.000 | |p|=\left|p e_{4}\right|=\left|\beta e_{4}\right|=\left|e_{4}\right|=1 | | |
| 0902.0431_FO0518 | 25 | 0.983 | p \in \operatorname{Sp}(1) | | |
| 0902.0431_FO0519 | 25 | 1.000 | \beta=\varphi(p, 1) | | |
| 0902.0431_FO0520 | 25 | 0.999 | \operatorname{Ker} \varphi=\{(1,1),(-1,-1)\}=\boldsymbol{Z}_{2} | | |
| 0902.0431_FO0521 | 25 | 0.802 | (S p(1) \times S p(1)) / \boldsymbol{Z}_{2} \cong\left(G_{2}\right)^{\gamma} | | |
| 0902.0431_FO0522 | 25 | 0.829 | (S p(1) \times S p(1)) / \boldsymbol{Z}_{2} \cong S O(4) | | |
| 0902.0431_FO0523 | 25 | 0.868 | f: \operatorname{Sp}(1) \times \operatorname{Sp}(1) \rightarrow \operatorname{SO}(4)=\operatorname{SO}(\boldsymbol{H}) | | |
| 0902.0431_FO0524 | 25 | 1.000 | \left(\mathfrak{g}_{2}\right)^{\gamma} | | |
| 0902.0431_FO0525 | 25 | 1.000 | \operatorname{dim}\left(\mathfrak{g}_{2}\right)^{\gamma}=6=3+3=\operatorname{dim}(\mathfrak{s p}(1) \oplus \mathfrak{s p}(1)) | | |
| 0902.0431_FO0526 | 26 | 1.000 | \alpha \in z\left(G_{2}\right) | | |
| 0902.0431_FO0527 | 26 | 1.000 | \gamma: \gamma \alpha=\alpha \gamma | | |
| 0902.0431_FO0528 | 26 | 0.913 | \alpha \in S O(4) | | |
| 0902.0431_FO0529 | 26 | 0.913 | \alpha \in z(S O(4) | | |
| 0902.0431_FO0530 | 26 | 1.000 | z(S O(4)) | | |
| 0902.0431_FO0531 | 26 | 1.000 | S O(4) | | |
| 0902.0431_FO0532 | 26 | 1.000 | \gamma \notin z\left(G_{2}\right) | | |
| 0902.0431_FO0533 | 26 | 1.000 | \alpha=1 | | |
| 0902.0431_FO0534 | 26 | 1.000 | \alpha \in G_{2}{ }^{C} | | |
| 0902.0431_FO0535 | 26 | 1.000 | \alpha x=(\alpha 1)(\alpha x) | | |
| 0902.0431_FO0536 | 26 | 1.000 | x \in \mathfrak{C}^{C} | | |
| 0902.0431_FO0537 | 26 | 1.000 | \left(\alpha e_{k}\right)\left(\alpha e_{k}\right)=\alpha\left(e_{k} e_{k}\right)=\alpha(-1)=-1 | | |
| 0902.0431_FO0538 | 26 | 1.000 | x=\alpha e_{k} | | |
| 0902.0431_FO0539 | 26 | 1.000 | N(x)=x \bar{x} \in C | | |
| 0902.0431_FO0540 | 26 | 1.000 | x x=-1 | | |
| 0902.0431_FO0541 | 26 | 1.000 | N(x) N(x)=1 | | |
| 0902.0431_FO0542 | 26 | 1.000 | N(x)= \pm 1 | | |
| 0902.0431_FO0543 | 26 | 1.000 | N(x)=-1 | | |
| 0902.0431_FO0544 | 26 | 1.000 | \bar{x}=-x x \bar{x}=-x N(x)=x | | |
| 0902.0431_FO0545 | 26 | 1.000 | x \in C | | |
| 0902.0431_FO0546 | 26 | 1.000 | x= \pm i | | |
| 0902.0431_FO0547 | 26 | 1.000 | \alpha\left(e_{k}\right)= \pm \alpha(i) | | |
| 0902.0431_FO0548 | 26 | 1.000 | e_{k}= \pm i | | |
| 0902.0431_FO0549 | 26 | 1.000 | N(x)=1 | | |
| 0902.0431_FO0550 | 26 | 1.000 | x \bar{x}=1 | | |
| 0902.0431_FO0551 | 26 | 1.000 | \bar{x}=-x x \bar{x}=-x N(x)=-x | | |
| 0902.0431_FO0552 | 26 | 1.000 | \langle x, y\rangle | | |
| 0902.0431_FO0553 | 27 | 1.000 | \alpha \in \operatorname{Hom}_{C}\left(\mathfrak{C}^{C}\right) | | |
| 0902.0431_FO0554 | 27 | 1.000 | \alpha^{*}:\left\langle\alpha^{*} x, y\right\rangle=\langle x, \alpha y\rangle | | |
| 0902.0431_FO0555 | 27 | 1.000 | \alpha^{*}=\tau \alpha^{-1} \tau \in G_{2}{ }^{C} | | |
| 0902.0431_FO0556 | 27 | 1.000 | \alpha^{C}: \mathfrak{C}^{C} \rightarrow \mathfrak{C}^{C} | | |
| 0902.0431_FO0557 | 27 | 0.998 | \alpha^{C} \in G_{2}{ }^{C} | | |
| 0902.0431_FO0558 | 27 | 0.998 | \alpha^{C} | | |
| 0902.0431_FO0559 | 27 | 0.998 | G_{2}{ }^{C}: G_{2} \subset G_{2}{ }^{C} | | |
| 0902.0431_FO0560 | 27 | 1.000 | \tau \alpha=\alpha \tau | | |
| 0902.0431_FO0561 | 27 | 1.000 | \left\langle\alpha^{*} x, y\right\rangle=\langle x, \alpha y\rangle=(\tau x, \alpha y)=\left(\alpha^{-1} \tau x, y\right)=\left\langle\tau \alpha^{-1} \tau x, y\right\rangle | | |
| 0902.0431_FO0562 | 27 | 1.000 | x, y \in \mathfrak{C}^{C} | | |
| 0902.0431_FO0563 | 27 | 1.000 | \tau \alpha x=\alpha \tau x=\alpha x | | |
| 0902.0431_FO0564 | 27 | 1.000 | \alpha x \in \mathfrak{C} | | |
| 0902.0431_FO0565 | 27 | 1.000 | \alpha^{\prime} | | |
| 0902.0431_FO0566 | 27 | 1.000 | \alpha^{\prime} \in G_{2} | | |
| 0902.0431_FO0567 | 27 | 1.000 | \alpha=\left(\alpha^{\prime}\right)^{C} | | |
| 0902.0431_FO0568 | 27 | 1.000 | \operatorname{Iso}_{C}\left(\mathfrak{C}^{C}\right)=G L(8, C) | | |
| 0902.0431_FO0569 | 27 | 1.000 | \alpha^{*} \in G_{2}{ }^{C} | | |
| 0902.0431_FO0570 | 27 | 1.000 | U\left(\mathfrak{C}^{C}\right)=\left\{\alpha \in \operatorname{Iso}_{C}\left(\mathfrak{C}^{C}\right) \mid\langle\alpha X, \alpha Y\rangle=\langle X, Y\rangle\right\} | | |
| 0902.0431_FO0571 | 27 | 1.000 | d=\operatorname{dim} G_{2}{ }^{C}-\operatorname{dim} G_{2}= | | |
| 0902.0431_FO0572 | 27 | 1.000 | 2 \times 14-14=14 | | |
| 0902.0431_FO0573 | 27 | 1.000 | \mathfrak{C}^{\prime}=\boldsymbol{H} \oplus \boldsymbol{H} e_{4}{ }^{\prime} | | |
| 0902.0431_FO0574 | 27 | 1.000 | \mathfrak{C}^{\prime} | | |
| 0902.0431_FO0575 | 27 | 1.000 | \left(\mathfrak{C}^{C}\right)_{\tau \gamma}=\{x \in | | |
| 0902.0431_FO0576 | 27 | 1.000 | \left.\mathfrak{C}^{C} \mid \tau \gamma x=x\right\} | | |
| 0902.0431_FO0577 | 28 | 1.000 | (x, y)^{\prime} | | |
| 0902.0431_FO0578 | 28 | 0.998 | (x, y)^{\prime}=\frac{1}{2}(x \bar{y}+y \bar{x}) | | |
| 0902.0431_FO0579 | 28 | 0.998 | (\gamma x, y)^{\prime} | | |
| 0902.0431_FO0580 | 28 | 0.986 | \alpha \in G_{2(2)} | | |
| 0902.0431_FO0581 | 28 | 0.986 | { }^{t} \alpha | | |
| 0902.0431_FO0582 | 28 | 0.858 | { }^{t} \alpha=\gamma \alpha^{-1} \gamma \in G_{2(2)} | | |
| 0902.0431_FO0583 | 28 | 1.000 | \operatorname{Iso}_{\boldsymbol{R}}\left(\mathfrak{C}^{\prime}\right)=G L(8, \boldsymbol{R}) | | |
| 0902.0431_FO0584 | 28 | 1.000 | O(8)=\left\{\alpha \in \operatorname{Iso}_{\boldsymbol{R}}\left(\mathfrak{C}^{\prime}\right) \mid(\alpha x, \alpha y)=(x, y)\right\} | | |
| 0902.0431_FO0585 | 28 | 0.999 | z\left(G_{2(2)}\right) | | |
| 0902.0431_FO0586 | 28 | 1.000 | a \in S^{n-1}=\left\{a \in \boldsymbol{R}^{n} \mid(a, a)=1\right\} | | |
| 0902.0431_FO0587 | 28 | 1.000 | D_{a} \in O(n)= | | |
| 0902.0431_FO0588 | 28 | 0.999 | O\left(\boldsymbol{R}^{n}\right)=\left\{A \in \operatorname{Iso}_{\boldsymbol{R}}\left(\boldsymbol{R}^{n}\right) \mid(A x, A y)=(x, y)\right\} | | |
| 0902.0431_FO0589 | 28 | 1.000 | D_{a} | | |
| 0902.0431_FO0590 | 28 | 0.570 | \operatorname{det}\left(D_{a}\right)=-1 | | |
| 0902.0431_FO0591 | 28 | 1.000 | O(n) | | |
| 0902.0431_FO0592 | 28 | 1.000 | A \in O(n) | | |
| 0902.0431_FO0593 | 28 | 1.000 | A \in S O(n) | | |
| 0902.0431_FO0594 | 29 | 0.999 | \alpha_{3} \in S O(8) | | |
| 0902.0431_FO0595 | 29 | 1.000 | \alpha_{1}, \alpha_{2} \in S O(8) | | |
| 0902.0431_FO0596 | 29 | 1.000 | \alpha_{1}, \alpha_{2} | | |
| 0902.0431_FO0597 | 29 | 1.000 | \alpha_{3} | | |
| 0902.0431_FO0598 | 29 | 1.000 | -\alpha_{1},-\alpha_{2} | | |
| 0902.0431_FO0599 | 29 | 1.000 | \alpha_{3}=D_{b} D_{a}, a, b \in \mathfrak{C},|a|=|b|=1 | | |
| 0902.0431_FO0600 | 29 | 1.000 | \alpha_{3} x=D_{b} D_{a} x=b(\bar{a} x \bar{a}) b | | |
| 0902.0431_FO0601 | 29 | 1.000 | \alpha_{1}, \alpha_{2}: \mathfrak{C} \rightarrow \mathfrak{C} | | |
| 0902.0431_FO0602 | 29 | 1.000 | \alpha_{1} x=b(\bar{a} x), \alpha_{2} x=(x \bar{a}) b | | |
| 0902.0431_FO0603 | 29 | 1.000 | \alpha_{3}=1 | | |
| 0902.0431_FO0604 | 29 | 1.000 | \alpha_{1} 1=p | | |
| 0902.0431_FO0605 | 29 | 1.000 | |p|=1 | | |
| 0902.0431_FO0606 | 29 | 1.000 | p\left(\alpha_{2} y\right)=y | | |
| 0902.0431_FO0607 | 29 | 1.000 | \alpha_{2} y=\bar{p} y | | |
| 0902.0431_FO0608 | 29 | 1.000 | \alpha_{1} x=x \bar{q} | | |
| 0902.0431_FO0609 | 29 | 1.000 | q=\alpha_{2} 1 | | |
| 0902.0431_FO0610 | 29 | 1.000 | (x \bar{q})(\bar{p} y)=x y | | |
| 0902.0431_FO0611 | 29 | 1.000 | \bar{q} \bar{p}=1 | | |
| 0902.0431_FO0612 | 29 | 1.000 | \bar{q}=p | | |
| 0902.0431_FO0613 | 29 | 1.000 | p y | | |
| 0902.0431_FO0614 | 29 | 1.000 | y | | |
| 0902.0431_FO0615 | 29 | 1.000 | p | | |
| 0902.0431_FO0616 | 29 | 1.000 | p= \pm 1 | | |
| 0902.0431_FO0617 | 29 | 1.000 | \alpha_{1}=\alpha_{2}=1 | | |
| 0902.0431_FO0618 | 29 | 1.000 | \alpha_{1}=\alpha_{2}=-1 | | |
| 0902.0431_FO0619 | 29 | 1.000 | \alpha_{1}, \alpha_{2}, \alpha_{3} \in O(8) | | |
| 0902.0431_FO0620 | 29 | 1.000 | x=0 | | |
| 0902.0431_FO0621 | 29 | 1.000 | y=0 | | |
| 0902.0431_FO0622 | 29 | 1.000 | x, y \neq 0 | | |
| 0902.0431_FO0623 | 29 | 1.000 | \overline{\alpha_{1} x} | | |
| 0902.0431_FO0624 | 29 | 1.000 | \alpha_{3}(\overline{x y}) | | |
| 0902.0431_FO0625 | 30 | 0.862 | \left.\left(\alpha_{1} x\right)\left(\alpha_{2} y\right)=\overline{\alpha_{3}(\overline{x y})}\right) | | |
| 0902.0431_FO0626 | 30 | 0.862 | |x|^{2}\left(\alpha_{2} y\right)\left(\alpha_{3}(\overline{x y})\right)=\overline{\alpha_{1} x}|x y|^{2} | | |
| 0902.0431_FO0627 | 30 | 1.000 | \overline{y z} | | |
| 0902.0431_FO0628 | 30 | 1.000 | \left(\alpha_{2} y\right)\left(\alpha_{3}(\bar{y} y z)\right)=\overline{\alpha_{1}(\overline{y z})}|y|^{2} | | |
| 0902.0431_FO0629 | 30 | 1.000 | \alpha_{1}, \alpha_{2}, \alpha_{3} \in S O(8) | | |
| 0902.0431_FO0630 | 30 | 1.000 | \alpha_{1} \notin S O(8) | | |
| 0902.0431_FO0631 | 30 | 1.000 | \epsilon: \mathfrak{C} \rightarrow \mathfrak{C} | | |
| 0902.0431_FO0632 | 30 | 1.000 | \epsilon x=\bar{x} | | |
| 0902.0431_FO0633 | 30 | 1.000 | O(8) | | |
| 0902.0431_FO0634 | 30 | 1.000 | \operatorname{det} \epsilon=-1 | | |
| 0902.0431_FO0635 | 30 | 1.000 | \beta_{1}=\epsilon \alpha_{1}^{-1} \in S O(8) | | |
| 0902.0431_FO0636 | 30 | 1.000 | \beta_{1} | | |
| 0902.0431_FO0637 | 30 | 0.995 | \beta_{2}, \beta_{3} \in S O(8) | | |
| 0902.0431_FO0638 | 30 | 1.000 | \beta_{2} \alpha_{2}=\gamma_{2} | | |
| 0902.0431_FO0639 | 30 | 1.000 | \beta_{3} \alpha_{3}=\gamma_{3} | | |
| 0902.0431_FO0640 | 30 | 1.000 | \gamma_{2} y=\gamma_{3} y | | |
| 0902.0431_FO0641 | 30 | 1.000 | \gamma_{2}=\gamma_{3} | | |
| 0902.0431_FO0642 | 30 | 1.000 | \gamma_{2} 1=p | | |
| 0902.0431_FO0643 | 30 | 1.000 | \bar{x} p=\gamma_{2} x | | |
| 0902.0431_FO0644 | 30 | 1.000 | y=p | | |
| 0902.0431_FO0645 | 30 | 1.000 | \bar{x}=(\overline{x p}) p | | |
| 0902.0431_FO0646 | 30 | 1.000 | \bar{x} \bar{p}=\overline{x p} | | |
| 0902.0431_FO0647 | 30 | 0.999 | \bar{x} \bar{y}=\overline{x y} | | |
| 0902.0431_FO0648 | 30 | 1.000 | \alpha_{1} \in S O(8) | | |
| 0902.0431_FO0649 | 30 | 1.000 | \alpha_{2}, \alpha_{3} | | |
| 0902.0431_FO0650 | 30 | 1.000 | \alpha_{2}, \alpha_{3} \in S O(8) | | |
| 0902.0431_FO0651 | 31 | 0.943 | |a|=1 | | |
| 0902.0431_FO0652 | 31 | 0.943 | \alpha_{a}: \mathfrak{C} \rightarrow \mathfrak{C} | | |
| 0902.0431_FO0653 | 31 | 1.000 | a^{3}= \pm 1 | | |
| 0902.0431_FO0654 | 31 | 1.000 | (\bar{a} x)(y \bar{a})=\bar{a}(x y) \bar{a} | | |
| 0902.0431_FO0655 | 31 | 1.000 | a x | | |
| 0902.0431_FO0656 | 31 | 1.000 | y a | | |
| 0902.0431_FO0657 | 31 | 1.000 | \alpha_{a} | | |
| 0902.0431_FO0658 | 31 | 1.000 | a^{2}= \pm \bar{a} | | |
| 0902.0431_FO0659 | 31 | 0.357 | \omega_{1}{ }^{3}=1 | | |
| 0902.0431_FO0660 | 31 | 0.357 | \alpha_{\bar{\omega}_{1}} \in G_{2} | | |
| 0902.0431_FO0661 | 31 | 1.000 | \alpha_{\bar{\omega}_{1}} | | |
| 0902.0431_FO0662 | 31 | 1.000 | \alpha_{\bar{\omega}_{1}}=w | | |
| 0902.0431_FO0663 | 31 | 0.991 | \alpha \in S O(7) | | |
| 0902.0431_FO0664 | 31 | 0.991 | \widetilde{\alpha}, \alpha^{\prime} \in S O(8) | | |
| 0902.0431_FO0665 | 31 | 1.000 | \widetilde{\alpha} y=\alpha^{\prime} y | | |
| 0902.0431_FO0666 | 31 | 1.000 | \widetilde{\alpha}=\alpha^{\prime} | | |
| 0902.0431_FO0667 | 31 | 1.000 | \alpha, \widetilde{\alpha} \in S O(8) | | |
| 0902.0431_FO0668 | 31 | 1.000 | (\alpha 1)(\widetilde{\alpha} y)= | | |
| 0902.0431_FO0669 | 31 | 0.986 | \widetilde{\alpha} y | | |
| 0902.0431_FO0670 | 31 | 1.000 | \widetilde{B}_{3} | | |
| 0902.0431_FO0671 | 32 | 0.671 | \quad \widetilde{B}_{3} / G_{2} \simeq S^{7}, \quad \widetilde{B}_{3} \cap S O(7)=G_{2} | | |
| 0902.0431_FO0672 | 32 | 1.000 | S^{7}=\{a \in \mathfrak{C}| | a \mid=1\} | | |
| 0902.0431_FO0673 | 32 | 1.000 | S^{7} | | |
| 0902.0431_FO0674 | 32 | 1.000 | b_{0} \in S^{7} | | |
| 0902.0431_FO0675 | 32 | 1.000 | 1 \in S^{7} | | |
| 0902.0431_FO0676 | 32 | 0.999 | \alpha \in \widetilde{B}_{3} | | |
| 0902.0431_FO0677 | 32 | 0.999 | a_{1} \in S^{7} | | |
| 0902.0431_FO0678 | 32 | 0.999 | \left(1, a_{1}\right)=0 | | |
| 0902.0431_FO0679 | 32 | 1.000 | a_{2} \in S^{7} | | |
| 0902.0431_FO0680 | 32 | 1.000 | \left(1, a_{2}\right)=\left(a_{1}, a_{2}\right)=0 | | |
| 0902.0431_FO0681 | 32 | 1.000 | a_{3} \in S^{7} | | |
| 0902.0431_FO0682 | 32 | 1.000 | a_{3} b_{0}=a_{2}\left(a_{1} b_{0}\right) | | |
| 0902.0431_FO0683 | 32 | 1.000 | a_{3}=\left(a_{2}\left(a_{1} b_{0}\right)\right) \bar{b}_{0} | | |
| 0902.0431_FO0684 | 32 | 1.000 | \left(1, a_{3}\right)=\left(a_{1}, a_{3}\right)=\left(a_{2}, a_{3}\right)=0 | | |
| 0902.0431_FO0685 | 32 | 1.000 | a_{4} \in S^{7} | | |
| 0902.0431_FO0686 | 32 | 1.000 | \left(1, a_{4}\right)=\left(a_{1}, a_{4}\right)=\left(a_{2}, a_{4}\right)=\left(a_{3}, a_{4}\right)=0 | | |
| 0902.0431_FO0687 | 32 | 1.000 | a_{5}, a_{6}, a_{7} \in S^{7} | | |
| 0902.0431_FO0688 | 32 | 1.000 | \left(a_{i}, a_{j}\right)=\delta_{i j}, i, j=0,1, \cdots, 7 | | |
| 0902.0431_FO0689 | 32 | 1.000 | \left\{e_{0}, e_{1}, \cdots, e_{7}\right\} | | |
| 0902.0431_FO0690 | 32 | 1.000 | \left\{a_{0}=1, a_{1}, \cdots, a_{7}\right\} | | |
| 0902.0431_FO0691 | 33 | 1.000 | \left\{b_{0}, b_{1}, \cdots, b_{7}\right\} | | |
| 0902.0431_FO0692 | 33 | 1.000 | \widetilde{\alpha} | | |
| 0902.0431_FO0693 | 33 | 1.000 | \mathfrak{C} \rightarrow \mathfrak{C} | | |
| 0902.0431_FO0694 | 33 | 1.000 | \widetilde{\alpha} x=(\alpha x) b_{0} | | |
| 0902.0431_FO0695 | 33 | 0.719 | \alpha \in O(7), \tilde{\alpha} \in O(8) | | |
| 0902.0431_FO0696 | 33 | 0.719 | \alpha \in S O(7), \tilde{\alpha} \in S O(8) | | |
| 0902.0431_FO0697 | 33 | 1.000 | \widetilde{\alpha} \in \widetilde{B}_{3} | | |
| 0902.0431_FO0698 | 33 | 1.000 | \widetilde{\alpha} 1=b_{0} | | |
| 0902.0431_FO0699 | 33 | 1.000 | \widetilde{\alpha}^{-1} b_{0}=1 | | |
| 0902.0431_FO0700 | 33 | 1.000 | \widetilde{\alpha} 1=1 | | |
| 0902.0431_FO0701 | 33 | 1.000 | \alpha=\widetilde{\alpha} | | |
| 0902.0431_FO0702 | 33 | 1.000 | \widetilde{\alpha} \in G_{2} | | |
| 0902.0431_FO0703 | 33 | 1.000 | \widetilde{B}_{3} / G_{2} \simeq S^{7} | | |
| 0902.0431_FO0704 | 33 | 1.000 | p: \widetilde{B_{3}} \rightarrow S O(7) | | |
| 0902.0431_FO0705 | 33 | 1.000 | p(\widetilde{\alpha})=\alpha | | |
| 0902.0431_FO0706 | 33 | 1.000 | \operatorname{Ker} p=\{1,-1\} | | |
| 0902.0431_FO0707 | 33 | 0.589 | \left.\alpha e_{1}=\left(\widetilde{\alpha} e_{2}\right)\left(\overline{\widetilde{\alpha} \bar{e}_{3}}\right)\right) | | |
| 0902.0431_FO0708 | 33 | 1.000 | \widetilde{D}_{4} | | |
| 0902.0431_FO0709 | 33 | 1.000 | S O(8) \times S O(8) \times S O(8) | | |
| 0902.0431_FO0710 | 33 | 0.996 | (\alpha, \widetilde{\alpha}, \kappa \widetilde{\alpha}) | | |
| 0902.0431_FO0711 | 33 | 0.996 | (\alpha x)(\widetilde{\alpha} y)=\widetilde{\alpha}(x y), x, y \in \mathfrak{C} | | |
| 0902.0431_FO0712 | 34 | 0.999 | a \in S^{7} | | |
| 0902.0431_FO0713 | 34 | 0.988 | \alpha_{1} 1=a | | |
| 0902.0431_FO0714 | 34 | 0.988 | \alpha_{1} | | |
| 0902.0431_FO0715 | 34 | 1.000 | \left(\alpha_{1}, \alpha_{2}, \alpha_{3}\right) \in \widetilde{D}_{4} | | |
| 0902.0431_FO0716 | 34 | 1.000 | \left(\alpha_{1}, \alpha_{2}, \alpha_{3}\right) 1=a | | |
| 0902.0431_FO0717 | 34 | 0.999 | \left(\alpha_{1}, \alpha_{2}, \alpha_{3}\right) 1=1 | | |
| 0902.0431_FO0718 | 34 | 0.999 | \alpha_{1} 1=1 | | |
| 0902.0431_FO0719 | 34 | 0.999 | \alpha_{1} \in S O(7) | | |
| 0902.0431_FO0720 | 34 | 1.000 | \left(\alpha_{1}, \alpha_{2}, \alpha_{3}\right) \in \operatorname{Spin}(7) | | |
| 0902.0431_FO0721 | 34 | 1.000 | \widetilde{D}_{4} / \operatorname{Spin}(7) | | |
| 0902.0431_FO0722 | 34 | 1.000 | \simeq S^{7} | | |
| 0902.0431_FO0723 | 34 | 1.000 | p: \widetilde{D}_{4} \rightarrow S O(8) | | |
| 0902.0431_FO0724 | 34 | 1.000 | \operatorname{Ker} p=\{(1,1,1),(1,-1,-1)\} | | |
| 0902.0431_FO0725 | 34 | 0.944 | \operatorname{SO}(8) | | |
| 0902.0431_FO0726 | 34 | 0.944 | z(\operatorname{Spin}(8)) | | |
| 0902.0431_FO0727 | 34 | 1.000 | \boldsymbol{Z}_{2} \times \boldsymbol{Z}_{2} | | |
| 0902.0431_FO0728 | 34 | 1.000 | z(S O(8))=\{1,-1\} | | |
| 0902.0431_FO0729 | 35 | 0.982 | \kappa, \pi, \nu: \operatorname{Spin}(8) \rightarrow \operatorname{Spin}(8) | | |
| 0902.0431_FO0730 | 35 | 1.000 | \kappa: S O(8) \rightarrow S O(8) | | |
| 0902.0431_FO0731 | 35 | 1.000 | (\kappa \alpha) x=\overline{\alpha \bar{x}}, x \in \mathfrak{C} | | |
| 0902.0431_FO0732 | 35 | 0.884 | \kappa, \pi | | |
| 0902.0431_FO0733 | 35 | 0.884 | \operatorname{Aut}(\operatorname{Spin}(8)) | | |
| 0902.0431_FO0734 | 35 | 1.000 | \left(\alpha_{1}, \alpha_{2}, \alpha_{3}\right) \in \operatorname{Spin}(8) | | |
| 0902.0431_FO0735 | 35 | 1.000 | \nu\left(\alpha_{1}, \alpha_{2}, \alpha_{3}\right)=\left(\alpha_{1}, \alpha_{2}, \alpha_{3}\right) | | |
| 0902.0431_FO0736 | 35 | 1.000 | \alpha_{1}= | | |
| 0902.0431_FO0737 | 35 | 1.000 | \alpha_{2}=\alpha_{3}(=\alpha) | | |
| 0902.0431_FO0738 | 35 | 1.000 | a=\alpha 1 | | |
| 0902.0431_FO0739 | 35 | 1.000 | a(\alpha y)=\kappa \alpha(y) | | |
| 0902.0431_FO0740 | 35 | 1.000 | (\alpha x) a=\kappa \alpha(x) | | |
| 0902.0431_FO0741 | 35 | 1.000 | a \in \boldsymbol{R} | | |
| 0902.0431_FO0742 | 35 | 1.000 | a= \pm 1 | | |
| 0902.0431_FO0743 | 35 | 1.000 | a=-1 | | |
| 0902.0431_FO0744 | 35 | 1.000 | (-1)(-1)=-1 | | |
| 0902.0431_FO0745 | 35 | 1.000 | a=1 | | |
| 0902.0431_FO0746 | 35 | 1.000 | \kappa \alpha=\alpha | | |
| 0902.0431_FO0747 | 35 | 1.000 | \nu^{2}: \operatorname{Spin}(8) \rightarrow \operatorname{Spin}(8) | | |
| 0902.0431_FO0748 | 35 | 0.997 | \nu^{2} | | |
| 0902.0431_FO0749 | 35 | 0.997 | S O(8) \cong S s(8) | | |
| 0902.0431_FO0750 | 36 | 0.999 | \mathfrak{J}=\mathfrak{J}(3, \mathfrak{C}) | | |
| 0902.0431_FO0751 | 36 | 0.999 | 3 \times 3 | | |
| 0902.0431_FO0752 | 36 | 0.998 | X^{*}={ }^{t} \bar{X} | | |
| 0902.0431_FO0753 | 36 | 0.998 | X \in \mathfrak{J} | | |
| 0902.0431_FO0754 | 36 | 1.000 | X \circ Y | | |
| 0902.0431_FO0755 | 36 | 0.964 | \operatorname{trace} \operatorname{tr}(X) | | |
| 0902.0431_FO0756 | 36 | 0.964 | (X, Y) | | |
| 0902.0431_FO0757 | 36 | 0.964 | \operatorname{tr}(X, Y, Z) | | |
| 0902.0431_FO0758 | 36 | 1.000 | X \times Y | | |
| 0902.0431_FO0759 | 36 | 0.839 | E | | |
| 0902.0431_FO0760 | 36 | 0.839 | (X, Y, Z) | | |
| 0902.0431_FO0761 | 36 | 1.000 | \operatorname{det} X | | |
| 0902.0431_FO0762 | 36 | 1.000 | X=X(\xi, x), Y=Y(\eta, y) | | |
| 0902.0431_FO0763 | 36 | 1.000 | Z=Z(\zeta, z) \in \mathfrak{J} | | |
| 0902.0431_FO0764 | 37 | 1.000 | X \circ Y=Y \circ X, \quad X \times Y=Y \times X | | |
| 0902.0431_FO0765 | 37 | 1.000 | E \circ X=X, \quad E \times X=\frac{1}{2}(\operatorname{tr}(X) E-X), \quad E \times E=E | | |
| 0902.0431_FO0766 | 37 | 0.990 | \operatorname{tr}(X, Y, Z)=\operatorname{tr}(Y, Z, X)=\operatorname{tr}(Z, X, Y)=\operatorname{tr}(X, Z, Y)=\operatorname{tr}(Y, X, Z) | | |
| 0902.0431_FO0767 | 37 | 0.971 | =\operatorname{tr}(Z, Y, X) | | |
| 0902.0431_FO0768 | 37 | 0.958 | \quad(X, E)=(X, E, E)=\operatorname{tr}(X, E, E)=\operatorname{tr}(X), \quad \operatorname{tr}(X, Y, E)=(X, Y) | | |
| 0902.0431_FO0769 | 37 | 0.772 | \quad \operatorname{tr}(X \times Y)=\frac{1}{2}(\operatorname{tr}(X) \operatorname{tr}(Y)-(X, Y)) | | |
| 0902.0431_FO0770 | 37 | 1.000 | (X \times X) \circ X=(\operatorname{det} X) E | | |
| 0902.0431_FO0771 | 37 | 1.000 | (X \times X) \times(X \times X)=(\operatorname{det} X) X | | |
| 0902.0431_FO0772 | 37 | 0.997 | (X, Y), \operatorname{tr}(X, Y, Z),(X, Y, Z) | | |
| 0902.0431_FO0773 | 38 | 0.673 | \bmod 3 | | |
| 0902.0431_FO0774 | 38 | 1.000 | \alpha \in F_{4} | | |
| 0902.0431_FO0775 | 38 | 1.000 | \alpha E=E | | |
| 0902.0431_FO0776 | 38 | 1.000 | \operatorname{tr}(\alpha X)=\operatorname{tr}(X), X \in \mathfrak{J} | | |
| 0902.0431_FO0777 | 38 | 1.000 | E \circ X=X | | |
| 0902.0431_FO0778 | 38 | 1.000 | \alpha E \circ \alpha X=\alpha X | | |
| 0902.0431_FO0779 | 38 | 1.000 | X=\alpha^{-1} E | | |
| 0902.0431_FO0780 | 38 | 1.000 | \alpha E \circ E=E | | |
| 0902.0431_FO0781 | 38 | 1.000 | X \circ(X \times X)=(\operatorname{det} X) E | | |
| 0902.0431_FO0782 | 38 | 1.000 | \alpha X | | |
| 0902.0431_FO0783 | 38 | 1.000 | \alpha^{-1} \in F_{4} | | |
| 0902.0431_FO0784 | 38 | 1.000 | X=F_{i}\left(e_{j}\right), i=1,2,3, j=0,1, \cdots, 7 | | |
| 0902.0431_FO0785 | 38 | 1.000 | \operatorname{tr}\left(\left(\alpha F_{i}\left(e_{j}\right)\right)^{2}\right)=2 | | |
| 0902.0431_FO0786 | 39 | 1.000 | i=1,2,3 | | |
| 0902.0431_FO0787 | 39 | 1.000 | \operatorname{tr}(\alpha X)=\operatorname{tr}(X) | | |
| 0902.0431_FO0788 | 39 | 1.000 | X=E_{i}, F_{i}\left(e_{j}\right) | | |
| 0902.0431_FO0789 | 39 | 1.000 | \alpha \in \operatorname{Hom}_{\boldsymbol{R}}(\mathfrak{J}) | | |
| 0902.0431_FO0790 | 39 | 1.000 | (X, Y):\left({ }^{t} \alpha X, Y\right)=(X, \alpha Y) | | |
| 0902.0431_FO0791 | 39 | 1.000 | \alpha \in \operatorname{Iso}_{\boldsymbol{R}}(\mathfrak{J}) | | |
| 0902.0431_FO0792 | 39 | 0.588 | \operatorname{det}(\alpha X)=\operatorname{det} X, \quad | | |
| 0902.0431_FO0793 | 39 | 0.999 | (\alpha X, \alpha Y, \alpha Z)=(X, Y, Z), \quad | | |
| 0902.0431_FO0794 | 39 | 0.999 | X, Y, Z \in \mathfrak{J} | | |
| 0902.0431_FO0795 | 39 | 1.000 | \alpha X \times \alpha Y={ }^{t} \alpha^{-1}(X \times Y), \quad | | |
| 0902.0431_FO0796 | 39 | 1.000 | X, Y \in \mathfrak{J} | | |
| 0902.0431_FO0797 | 39 | 0.997 | \alpha X \times \alpha X={ }^{t} \alpha^{-1}(X \times X), \quad | | |
| 0902.0431_FO0798 | 39 | 0.992 | \operatorname{det}(\alpha X)=\operatorname{det} X | | |
| 0902.0431_FO0799 | 39 | 0.992 | (\alpha X, \alpha X, \alpha X)=(X, X, X) | | |
| 0902.0431_FO0800 | 39 | 1.000 | \lambda X+\mu Y+\nu Z | | |
| 0902.0431_FO0801 | 39 | 1.000 | \lambda \mu \nu | | |
| 0902.0431_FO0802 | 39 | 1.000 | \lambda X+\mu Y | | |
| 0902.0431_FO0803 | 39 | 1.000 | \lambda \mu | | |
| 0902.0431_FO0804 | 39 | 1.000 | Y=X \times X, X \in \mathfrak{J} | | |
| 0902.0431_FO0805 | 39 | 1.000 | \operatorname{det} X \neq 0 | | |
| 0902.0431_FO0806 | 39 | 1.000 | { }^{t} \alpha^{-1} X \times{ }^{t} \alpha^{-1} X=\alpha(X \times X) | | |
| 0902.0431_FO0807 | 39 | 1.000 | \operatorname{det}\left({ }^{t} \alpha^{-1} X\right)=\operatorname{det} X | | |
| 0902.0431_FO0808 | 39 | 1.000 | \alpha^{-1} | | |
| 0902.0431_FO0809 | 39 | 1.000 | \operatorname{det}\left({ }^{t} \alpha X\right)=\operatorname{det} X | | |
| 0902.0431_FO0810 | 40 | 1.000 | \operatorname{det} X=0 | | |
| 0902.0431_FO0811 | 40 | 1.000 | \operatorname{det}\left({ }^{t} \alpha^{-1} X\right) \neq 0 | | |
| 0902.0431_FO0812 | 40 | 1.000 | { }^{t} \alpha X | | |
| 0902.0431_FO0813 | 40 | 0.670 | \left.\operatorname{det}{ }^{t} \alpha^{-1}\left({ }^{t} \alpha X\right)\right)=\operatorname{det}\left({ }^{t} \alpha X\right) | | |
| 0902.0431_FO0814 | 40 | 0.817 | \left.0=\operatorname{det} X=\operatorname{det}^{t} \alpha X\right) \neq 0 | | |
| 0902.0431_FO0815 | 40 | 0.817 | \operatorname{det}\left({ }^{t} \alpha^{-1} X\right)= | | |
| 0902.0431_FO0816 | 40 | 1.000 | \operatorname{det}\left({ }^{t} \alpha X\right)=0 | | |
| 0902.0431_FO0817 | 40 | 1.000 | \operatorname{det}\left({ }^{t} \alpha^{-1} X\right)=\operatorname{det}\left({ }^{t} \alpha X\right)=\operatorname{det} X | | |
| 0902.0431_FO0818 | 40 | 0.924 | \Rightarrow(2)(\alpha X, \alpha Y)=\operatorname{tr}(\alpha X \circ \alpha Y)=\operatorname{tr}(\alpha(X \circ Y))=\operatorname{tr}(X \circ Y) | | |
| 0902.0431_FO0819 | 40 | 0.521 | =(X, Y) | | |
| 0902.0431_FO0820 | 40 | 0.521 | \operatorname{tr}(\alpha X, \alpha Y, \alpha Z)=(\alpha X, \alpha Y \circ \alpha Z)=(\alpha X, \alpha(Y \circ Z))=(X, Y \circ | | |
| 0902.0431_FO0821 | 40 | 1.000 | Z)=\operatorname{tr}(X, Y, Z) | | |
| 0902.0431_FO0822 | 40 | 0.699 | \Rightarrow(1)(\alpha X \circ \alpha Y, \alpha Z)=\operatorname{tr}(\alpha X, \alpha Y, \alpha Z)=\operatorname{tr}(X, Y, Z)=(X \circ Y, Z)=(\alpha(X \circ | | |
| 0902.0431_FO0823 | 40 | 0.998 | Y), \alpha Z) | | |
| 0902.0431_FO0824 | 40 | 0.998 | \alpha Z | | |
| 0902.0431_FO0825 | 40 | 0.998 | \alpha X \circ \alpha Y=\alpha(X \circ Y) | | |
| 0902.0431_FO0826 | 40 | 0.946 | \Rightarrow(5)(\alpha(X \times Y), \alpha Z)=(X \times Y, Z)=(X, Y, Z)=(\alpha X, \alpha Y, \alpha Z) | | |
| 0902.0431_FO0827 | 40 | 1.000 | =(\alpha X \times \alpha Y, \alpha Z) | | |
| 0902.0431_FO0828 | 40 | 1.000 | \alpha Z \in \mathfrak{J} | | |
| 0902.0431_FO0829 | 40 | 1.000 | \alpha X \times \alpha Y=\alpha(X \times Y) | | |
| 0902.0431_FO0831 | 40 | 1.000 | \alpha E=P | | |
| 0902.0431_FO0832 | 40 | 1.000 | X=\alpha^{-1} E_{1} | | |
| 0902.0431_FO0833 | 40 | 1.000 | P=\rho_{1} E_{1}+\rho_{2} E_{2}+\rho_{3} E_{3}+F_{1}\left(p_{1}\right)+F_{2}\left(p_{2}\right)+F_{3}\left(p_{3}\right) | | |
| 0902.0431_FO0834 | 40 | 1.000 | \lambda=\operatorname{tr}\left(\alpha^{-1} E_{1}\right) | | |
| 0902.0431_FO0835 | 41 | 1.000 | p_{2}=p_{3}=0 | | |
| 0902.0431_FO0836 | 41 | 1.000 | X=\alpha^{-1} E_{2} | | |
| 0902.0431_FO0837 | 41 | 1.000 | p_{1}=0 | | |
| 0902.0431_FO0838 | 41 | 1.000 | X=\alpha^{-1} F_{1}(1) | | |
| 0902.0431_FO0839 | 41 | 1.000 | \mu=\operatorname{tr}\left(\alpha^{-1} F_{1}(1)\right) | | |
| 0902.0431_FO0840 | 41 | 1.000 | F_{1} | | |
| 0902.0431_FO0841 | 41 | 1.000 | \rho_{1}=1 | | |
| 0902.0431_FO0842 | 41 | 1.000 | \rho_{2}=\rho_{3}=1 | | |
| 0902.0431_FO0843 | 41 | 1.000 | (X, Y, Z)=\operatorname{tr}(X, Y, Z)-\frac{1}{2} \operatorname{tr}(X)(Y, Z)-\frac{1}{2} \operatorname{tr}(Y)(Z, X)- | | |
| 0902.0431_FO0844 | 41 | 0.992 | \frac{1}{2} \operatorname{tr}(Z)(X, Y)+\frac{1}{2} \operatorname{tr}(X) \operatorname{tr}(Y) \operatorname{tr}(Z) | | |
| 0902.0431_FO0845 | 41 | 0.992 | (\alpha X, \alpha Y, \alpha Z)=(X, Y, Z) | | |
| 0902.0431_FO0846 | 41 | 1.000 | \widetilde{\alpha}: \mathfrak{J} \rightarrow \mathfrak{J} | | |
| 0902.0431_FO0847 | 41 | 0.941 | \widetilde{\alpha} \in F_{4} | | |
| 0902.0431_FO0848 | 41 | 0.941 | \widetilde{\alpha} \in F_{4}: G_{2} \subset F_{4} | | |
| 0902.0431_FO0849 | 41 | 1.000 | \mathfrak{e}_{6(-26)} | | |
| 0902.0431_FO0850 | 42 | 1.000 | \phi \in \operatorname{Hom}_{\boldsymbol{R}}(\mathfrak{J}) | | |
| 0902.0431_FO0851 | 42 | 1.000 | ((\exp t \phi) X,(\exp t \phi) X,(\exp t \phi) X)= | | |
| 0902.0431_FO0852 | 42 | 1.000 | (X, X, X) | | |
| 0902.0431_FO0853 | 42 | 1.000 | t \in \boldsymbol{R} | | |
| 0902.0431_FO0854 | 42 | 1.000 | (\phi X, X, X)=0 | | |
| 0902.0431_FO0855 | 42 | 0.998 | -{ }^{t} \phi(X \times Y)=\phi X \times | | |
| 0902.0431_FO0856 | 42 | 1.000 | Y+X \times \phi Y | | |
| 0902.0431_FO0857 | 42 | 1.000 | \alpha=\exp t \phi | | |
| 0902.0431_FO0858 | 42 | 1.000 | { }^{t} \alpha^{-1}(X \times Y)=\alpha X \times \alpha Y | | |
| 0902.0431_FO0859 | 42 | 1.000 | \mathfrak{M}^{-} | | |
| 0902.0431_FO0860 | 42 | 1.000 | X, Y \in M(3, \mathfrak{C}) | | |
| 0902.0431_FO0861 | 42 | 1.000 | [X, Y] \in M(3, \mathfrak{C}) | | |
| 0902.0431_FO0862 | 42 | 0.997 | \quad\left[\mathfrak{M}^{-}, \mathfrak{J}\right] \subset \mathfrak{J}, \quad[\mathfrak{J}, \mathfrak{J}] \subset \mathfrak{M}^{-} | | |
| 0902.0431_FO0863 | 42 | 0.998 | \left[\mathfrak{M}^{-}, \mathfrak{J}\right] \subset \mathfrak{J} | | |
| 0902.0431_FO0864 | 42 | 0.998 | A \in \mathfrak{M}^{-} | | |
| 0902.0431_FO0865 | 42 | 0.998 | \widetilde{A}: \mathfrak{J} \rightarrow \mathfrak{J} | | |
| 0902.0431_FO0866 | 42 | 0.995 | X=\left(x_{i j}\right), x_{i j} \in \mathfrak{C}, \bar{x}_{i j}=x_{j i} | | |
| 0902.0431_FO0867 | 42 | 0.995 | (i, j) | | |
| 0902.0431_FO0868 | 42 | 0.995 | a_{i j} | | |
| 0902.0431_FO0869 | 42 | 0.995 | [X, X X]= | | |
| 0902.0431_FO0870 | 42 | 1.000 | X(X X)-(X X) X | | |
| 0902.0431_FO0871 | 43 | 0.996 | x_{i i} | | |
| 0902.0431_FO0872 | 43 | 0.996 | i \neq j | | |
| 0902.0431_FO0873 | 43 | 1.000 | a(x a)-(a x) a, a(\bar{a} x)-(a \bar{a}) x | | |
| 0902.0431_FO0874 | 43 | 1.000 | a_{i j}=0 | | |
| 0902.0431_FO0875 | 43 | 1.000 | i=j | | |
| 0902.0431_FO0876 | 43 | 1.000 | a_{11}=a_{22}=a_{33}(=a) | | |
| 0902.0431_FO0877 | 43 | 1.000 | [X, X X]=a E | | |
| 0902.0431_FO0878 | 43 | 1.000 | X, X X \in \mathfrak{J} | | |
| 0902.0431_FO0879 | 43 | 1.000 | [X, X X] \in \mathfrak{M}^{-} | | |
| 0902.0431_FO0880 | 43 | 1.000 | (a E)^{*}=-a E | | |
| 0902.0431_FO0881 | 43 | 1.000 | \bar{a} E=-a E | | |
| 0902.0431_FO0882 | 43 | 1.000 | \bar{a}=-a | | |
| 0902.0431_FO0883 | 43 | 1.000 | M(3, \mathfrak{C}) | | |
| 0902.0431_FO0884 | 43 | 1.000 | X=\left(x_{i j}\right), Y=\left(y_{i j}\right), Z=\left(z_{i j}\right) | | |
| 0902.0431_FO0885 | 43 | 0.997 | A \in \mathfrak{M}^{-}, \operatorname{tr}(A)=0 | | |
| 0902.0431_FO0886 | 43 | 0.997 | \widetilde{A} \in \mathfrak{f}_{4} | | |
| 0902.0431_FO0887 | 43 | 1.000 | =(A X, Y)-(X A, Y)+(X, A Y)-(X, Y A)=0 | | |
| 0902.0431_FO0888 | 43 | 1.000 | [X, X X]=a E, a \in \mathfrak{C}_{0} | | |
| 0902.0431_FO0889 | 44 | 1.000 | \lambda X+\mu X+\nu Z | | |
| 0902.0431_FO0890 | 44 | 1.000 | \widetilde{A}_{i}(a) \in \mathfrak{f}_{4} | | |
| 0902.0431_FO0891 | 44 | 1.000 | \widetilde{A}_{i}(a) | | |
| 0902.0431_FO0892 | 44 | 1.000 | \mathfrak{D}_{4}=\mathfrak{s o}(8) | | |
| 0902.0431_FO0893 | 45 | 1.000 | \varphi_{*}: \mathfrak{D}_{4} \rightarrow \mathfrak{d}_{4} | | |
| 0902.0431_FO0894 | 45 | 1.000 | \varphi_{*}\left(D_{1}\right)=\delta | | |
| 0902.0431_FO0895 | 45 | 1.000 | \delta \in \mathfrak{d}_{4} \subset \mathfrak{f}_{4} | | |
| 0902.0431_FO0896 | 45 | 1.000 | \delta \in \mathfrak{f}_{4} . \delta E_{i}=0, i=1,2,3 | | |
| 0902.0431_FO0897 | 45 | 1.000 | \delta \in \mathfrak{d}_{4} | | |
| 0902.0431_FO0898 | 45 | 1.000 | \delta X \in \mathfrak{J}_{i} | | |
| 0902.0431_FO0899 | 45 | 1.000 | X \in \mathfrak{J}_{i}, \delta | | |
| 0902.0431_FO0900 | 45 | 1.000 | \delta: \mathfrak{J}_{i} \rightarrow \mathfrak{J}_{i} | | |
| 0902.0431_FO0901 | 45 | 1.000 | D_{i}: \mathfrak{C} \rightarrow \mathfrak{C} | | |
| 0902.0431_FO0902 | 45 | 1.000 | \delta | | |
| 0902.0431_FO0903 | 45 | 1.000 | F_{i}(x) \circ F_{i}(y)=(x, y)\left(E_{i+1}+E_{i+2}\right) | | |
| 0902.0431_FO0904 | 45 | 1.000 | F_{i}\left(D_{i} x\right) \circ | | |
| 0902.0431_FO0905 | 45 | 1.000 | F_{i}(y)+F_{i}(x) \circ F_{i}\left(D_{i} y\right)=0 | | |
| 0902.0431_FO0906 | 45 | 1.000 | D_{i} \in \mathfrak{D}_{4}, i=1,2,3 | | |
| 0902.0431_FO0907 | 45 | 1.000 | F_{1}(x) \circ F_{2}(y)=\frac{1}{2} F_{3}(\overline{x y}) | | |
| 0902.0431_FO0908 | 45 | 1.000 | \delta \in \mathfrak{f}_{4} | | |
| 0902.0431_FO0909 | 45 | 0.988 | \operatorname{diag} A=0 | | |
| 0902.0431_FO0910 | 45 | 0.988 | a_{i i} | | |
| 0902.0431_FO0911 | 45 | 1.000 | E_{i} \circ E_{i}=E_{i} | | |
| 0902.0431_FO0912 | 45 | 1.000 | E_{i} \circ E_{j}=0, i \neq j | | |
| 0902.0431_FO0913 | 46 | 1.000 | \delta E_{i} | | |
| 0902.0431_FO0914 | 46 | 1.000 | a_{i} \in \mathfrak{C} | | |
| 0902.0431_FO0915 | 46 | 1.000 | A=2\left(\begin{array}{ccc}0 & a_{3} & -\bar{a}_{2} \\ -\bar{a}_{3} & 0 & a_{1} \\ a_{2} & -\bar{a}_{1} & 0\end{array}\right) | | |
| 0902.0431_FO0916 | 46 | 1.000 | a_{i} | | |
| 0902.0431_FO0917 | 46 | 1.000 | D=\delta-\widetilde{A} | | |
| 0902.0431_FO0918 | 46 | 1.000 | D E_{i}=0, i=1,2,3 | | |
| 0902.0431_FO0919 | 46 | 1.000 | D \in \mathfrak{d}_{4} | | |
| 0902.0431_FO0920 | 46 | 0.546 | \delta=D+\widetilde{A} | | |
| 0902.0431_FO0921 | 46 | 0.546 | D \in \mathfrak{d}_{4}, A \in \mathfrak{M}^{-}, \operatorname{diag} A=0 | | |
| 0902.0431_FO0922 | 46 | 1.000 | E_{i} | | |
| 0902.0431_FO0923 | 46 | 1.000 | \widetilde{A} E_{i}=0, i=1,2,3 | | |
| 0902.0431_FO0924 | 46 | 1.000 | A=0 | | |
| 0902.0431_FO0925 | 46 | 0.999 | \operatorname{dim} \mathfrak{f}_{4}=28+24=52 | | |
| 0902.0431_FO0926 | 46 | 1.000 | \mathfrak{J}^{C}=\left\{X_{1}+i X_{2} \mid X_{1}, X_{2} \in \mathfrak{J}\right\} | | |
| 0902.0431_FO0927 | 46 | 1.000 | \mathfrak{J}^{C} | | |
| 0902.0431_FO0928 | 46 | 0.991 | \operatorname{tr}(X, Y, Z),(X, Y, Z) | | |
| 0902.0431_FO0929 | 46 | 0.996 | \mathfrak{J} . \mathfrak{J}^{C} | | |
| 0902.0431_FO0930 | 46 | 1.000 | f_{4} | | |
| 0902.0431_FO0931 | 47 | 0.982 | A \in \mathfrak{J}^{C} | | |
| 0902.0431_FO0932 | 47 | 0.982 | \widetilde{A}: \mathfrak{J}^{C} \rightarrow \mathfrak{J}^{C} | | |
| 0902.0431_FO0933 | 47 | 0.996 | A \in \mathfrak{J}^{C}, \operatorname{tr}(A)=0 | | |
| 0902.0431_FO0934 | 47 | 0.996 | \widetilde{A} \in \mathfrak{e}_{6}{ }^{C} | | |
| 0902.0431_FO0935 | 47 | 1.000 | A, B \in \mathfrak{J}^{C} | | |
| 0902.0431_FO0936 | 47 | 1.000 | [\widetilde{A}, \widetilde{B}] \in \mathfrak{f}_{4}{ }^{C} | | |
| 0902.0431_FO0937 | 47 | 1.000 | (\widetilde{A} X, X, X)=(A \circ X, X \times X)=(A, X \circ(X \times X)) | | |
| 0902.0431_FO0938 | 47 | 1.000 | \mathfrak{d}_{4}{ }^{C} | | |
| 0902.0431_FO0939 | 47 | 0.997 | \mathfrak{d}_{4}{ }^{C} \mathfrak{C}^{C}=\left\{\sum_{i} D_{i} a_{i} \mid D_{i} \in \mathfrak{d}_{4}{ }^{C}, a_{i} \in \mathfrak{C}^{C}\right\}=\mathfrak{C}^{C} | | |
| 0902.0431_FO0940 | 47 | 1.000 | 0 \neq x=\sum_{i=0}^{7} x_{i} e_{i}, x_{i} \in C | | |
| 0902.0431_FO0941 | 47 | 1.000 | x_{i} \neq 0 | | |
| 0902.0431_FO0942 | 47 | 1.000 | x_{0} \neq 0 | | |
| 0902.0431_FO0943 | 47 | 1.000 | G_{i 0} | | |
| 0902.0431_FO0944 | 47 | 1.000 | W \neq\{0\} | | |
| 0902.0431_FO0945 | 47 | 1.000 | e_{0}=1 \in W | | |
| 0902.0431_FO0946 | 47 | 1.000 | G_{i 0} e_{0}=e_{i} | | |
| 0902.0431_FO0947 | 47 | 1.000 | W=\mathfrak{C}^{C} | | |
| 0902.0431_FO0948 | 48 | 1.000 | \mathfrak{d}_{4}{ }^{C} \mathfrak{C}^{C} | | |
| 0902.0431_FO0949 | 48 | 0.985 | \mathfrak{d}_{4}{ }^{C} \mathfrak{C}^{C}=\mathfrak{C}^{C} | | |
| 0902.0431_FO0950 | 48 | 1.000 | \delta \in \mathfrak{f}_{4}{ }^{C} | | |
| 0902.0431_FO0951 | 48 | 1.000 | A_{i}\left(a_{i}\right) \in\left(\mathfrak{M}^{-}\right)^{C} | | |
| 0902.0431_FO0952 | 48 | 1.000 | \widetilde{\mathfrak{A}}_{i}^{C}=\left\{\widetilde{A}_{i}(a) \mid a \in \mathfrak{C}^{C}\right\} | | |
| 0902.0431_FO0953 | 48 | 1.000 | \widetilde{\mathfrak{A}}^{C}=\widetilde{\mathfrak{A}}_{1}^{C} \oplus \widetilde{\mathfrak{A}}_{2}^{C} \oplus \widetilde{\mathfrak{A}}_{3}^{C} | | |
| 0902.0431_FO0954 | 48 | 0.998 | p: \mathfrak{f}_{4}{ }^{C} \rightarrow \mathfrak{d}_{4}{ }^{C} | | |
| 0902.0431_FO0955 | 48 | 0.998 | q: \mathfrak{f}_{4}{ }^{C} \rightarrow \widetilde{\mathfrak{A}}^{C} | | |
| 0902.0431_FO0956 | 48 | 0.998 | \mathfrak{f}_{4}{ }^{C}=\mathfrak{d}_{4}{ }^{C} \oplus \widetilde{\mathfrak{A}}^{C} | | |
| 0902.0431_FO0957 | 48 | 1.000 | a_{i} \in \mathfrak{C}^{C}, i=1,2,3 | | |
| 0902.0431_FO0958 | 48 | 1.000 | D+\sum_{i=1}^{3} \widetilde{A}_{i}\left(a_{i}\right) \in \mathfrak{a} | | |
| 0902.0431_FO0959 | 48 | 1.000 | D^{\prime} \in \mathfrak{d}_{4}{ }^{C} | | |
| 0902.0431_FO0960 | 48 | 1.000 | \mathfrak{d}_{4}{ }^{C} \cap \mathfrak{a} \neq\{0\} | | |
| 0902.0431_FO0961 | 48 | 1.000 | \widetilde{\mathfrak{A}}^{C} \cap \mathfrak{a} \neq\{0\} | | |
| 0902.0431_FO0962 | 48 | 1.000 | \mathfrak{d}_{4}{ }^{C} \cap \mathfrak{a}=\{0\} | | |
| 0902.0431_FO0963 | 48 | 1.000 | \widetilde{\mathfrak{A}}^{C} \cap \mathfrak{a}=\{0\} | | |
| 0902.0431_FO0964 | 48 | 1.000 | p \mid \mathfrak{a}: \mathfrak{a} \rightarrow \mathfrak{d}_{4}{ }^{C} | | |
| 0902.0431_FO0965 | 48 | 1.000 | \mathfrak{A}^{C} \cap \mathfrak{a}=\{0\} | | |
| 0902.0431_FO0966 | 48 | 1.000 | p(\mathfrak{a})=\mathfrak{d}_{4}{ }^{C} | | |
| 0902.0431_FO0967 | 48 | 1.000 | \operatorname{dim}_{C} \mathfrak{a}=\operatorname{dim}_{C} p(\mathfrak{a})=\operatorname{dim}_{C} \mathfrak{d}_{4}{ }^{C}=28 | | |
| 0902.0431_FO0968 | 48 | 1.000 | q \mid \mathfrak{a}: \mathfrak{a} \rightarrow \widetilde{\mathfrak{A}}^{C} | | |
| 0902.0431_FO0969 | 48 | 1.000 | \operatorname{dim}_{C} \mathfrak{a} \leq \operatorname{dim}_{C} \widetilde{\mathfrak{A}}^{C}=8 \times 3=24 | | |
| 0902.0431_FO0970 | 48 | 1.000 | \mathfrak{d}_{4}{ }^{C} \cap \mathfrak{a}=\mathfrak{d}_{4}{ }^{C} | | |
| 0902.0431_FO0971 | 48 | 1.000 | \mathfrak{a} \supset \mathfrak{d}_{4}{ }^{C} | | |
| 0902.0431_FO0972 | 48 | 0.991 | \left[D, \widetilde{A}_{i}\left(a_{i}\right)\right]=\widetilde{A}_{i}\left(D a_{i}\right) | | |
| 0902.0431_FO0973 | 48 | 0.874 | \mathfrak{a} \supset \mathfrak{d}_{4}{ }^{C} \oplus \widetilde{\mathfrak{A}}_{1}{ }^{C} \oplus \widetilde{\mathfrak{A}}_{2}{ }^{C} \oplus \widetilde{\mathfrak{A}}_{3}{ }^{C}=\mathfrak{f}_{4}{ }^{C} | | |
| 0902.0431_FO0974 | 48 | 1.000 | a_{1} \neq 0 | | |
| 0902.0431_FO0975 | 49 | 1.000 | b=c=0 | | |
| 0902.0431_FO0976 | 49 | 1.000 | \widetilde{A}_{1}(1) \in \mathfrak{a} | | |
| 0902.0431_FO0977 | 49 | 0.645 | \left[\widetilde{A}_{1}(1), \widetilde{A}_{1}\left(e_{1}\right)\right] F_{2}(1)=-2 F_{2}\left(e_{1}\right) | | |
| 0902.0431_FO0978 | 49 | 1.000 | b \neq 0 | | |
| 0902.0431_FO0979 | 49 | 1.000 | \widetilde{A}_{1}(1) | | |
| 0902.0431_FO0980 | 49 | 1.000 | c^{\prime}=0 | | |
| 0902.0431_FO0981 | 49 | 1.000 | c^{\prime} \neq 0 | | |
| 0902.0431_FO0982 | 49 | 1.000 | \mathfrak{a}=\mathfrak{f}_{4}{ }^{C} | | |
| 0902.0431_FO0983 | 49 | 0.821 | \delta=\sum_{i}\left[\widetilde{A}_{i}, \widetilde{B}_{i}\right], A_{i}, B_{i} \in \mathfrak{J}^{C} | | |
| 0902.0431_FO0984 | 49 | 1.000 | [\delta,[\widetilde{A}, \widetilde{B}]] X=\delta[\widetilde{A}, \widetilde{B}] X-[\widetilde{A}, \widetilde{B}] \delta X | | |
| 0902.0431_FO0985 | 49 | 0.994 | \mathfrak{a}=\left\{\sum_{i}\left[\widetilde{A}_{i}, \widetilde{B}_{i}\right] \mid A_{i}, B_{i} \in \mathfrak{J}^{C}\right\} | | |
| 0902.0431_FO0986 | 49 | 0.999 | \mathfrak{J}_{0}{ }^{C}=\left\{X \in \mathfrak{J}^{C} \mid \operatorname{tr}(X)=0\right\} | | |
| 0902.0431_FO0987 | 49 | 0.952 | \mathfrak{f}_{4}{ }^{C} \mathfrak{J}_{0}{ }^{C}=\left\{\sum_{i} \delta_{i} B_{i} \mid \delta_{i} \in \mathfrak{f}_{4}{ }^{C}, B_{i} \in \mathfrak{J}_{0}{ }^{C}\right\}=\mathfrak{J}_{0}{ }^{C} | | |
| 0902.0431_FO0988 | 49 | 0.964 | \mathfrak{J}_{0}{ }^{C} | | |
| 0902.0431_FO0989 | 49 | 0.964 | \mathfrak{f}_{4}{ }^{C}-C | | |
| 0902.0431_FO0990 | 49 | 1.000 | F_{i}(x) \neq 0(i=1,2,3) | | |
| 0902.0431_FO0991 | 49 | 1.000 | W=\mathfrak{J}_{0}{ }^{C} | | |
| 0902.0431_FO0992 | 49 | 1.000 | F_{i}(1) \in W | | |
| 0902.0431_FO0993 | 49 | 1.000 | \widetilde{A}_{i}(a) F_{i+2}(1)=F_{i+1}(\bar{a}) | | |
| 0902.0431_FO0994 | 49 | 1.000 | E_{i}-E_{i+1}, F_{i}(a) \in W | | |
| 0902.0431_FO0995 | 50 | 1.000 | y_{2}=y_{3}=0 | | |
| 0902.0431_FO0996 | 50 | 1.000 | F_{1}(1) \in W | | |
| 0902.0431_FO0997 | 50 | 1.000 | y_{2} \neq | | |
| 0902.0431_FO0998 | 50 | 0.996 | a \in \mathfrak{C}^{C}, a \neq 0 | | |
| 0902.0431_FO0999 | 50 | 0.996 | \left(y_{2}, a\right)=0 | | |
| 0902.0431_FO1000 | 50 | 0.996 | \widetilde{A}_{2}(a) | | |
| 0902.0431_FO1001 | 50 | 1.000 | -F_{3}(\bar{a})+F_{1}\left(\overline{a y_{3}}\right) \in W | | |
| 0902.0431_FO1002 | 50 | 1.000 | z_{1}=0 | | |
| 0902.0431_FO1003 | 50 | 1.000 | z_{1} \neq 0 | | |
| 0902.0431_FO1004 | 50 | 1.000 | b \in \mathfrak{C}^{C}, b \neq 0 | | |
| 0902.0431_FO1005 | 50 | 0.941 | \left(z_{1}, b\right)=0 | | |
| 0902.0431_FO1006 | 50 | 0.941 | \widetilde{A}_{1}(b) | | |
| 0902.0431_FO1007 | 50 | 0.941 | F_{2}(\bar{b}) \in W | | |
| 0902.0431_FO1008 | 50 | 1.000 | x_{2} \neq 0 | | |
| 0902.0431_FO1009 | 50 | 1.000 | x_{3} \neq 0 | | |
| 0902.0431_FO1010 | 50 | 1.000 | x_{1}=x_{2}=x_{3}=0 | | |
| 0902.0431_FO1011 | 50 | 1.000 | F_{1}(\xi-2 \eta) \in W | | |
| 0902.0431_FO1012 | 50 | 1.000 | \xi-2 \eta \neq 0 | | |
| 0902.0431_FO1013 | 50 | 1.000 | \xi-2 \eta=0 | | |
| 0902.0431_FO1014 | 50 | 1.000 | \widetilde{A}_{3}(1) | | |
| 0902.0431_FO1015 | 50 | 1.000 | F_{3}(3 \eta) \in W | | |
| 0902.0431_FO1016 | 50 | 1.000 | \mathfrak{f}_{4}{ }^{C} \mathfrak{J}_{0}{ }^{C} | | |
| 0902.0431_FO1017 | 50 | 1.000 | \mathfrak{f}_{4}{ }^{C} \mathfrak{J}_{0}{ }^{C}=\mathfrak{J}_{0}{ }^{C} | | |
| 0902.0431_FO1018 | 50 | 0.997 | A, B, C, D \in \mathfrak{J}^{C} | | |
| 0902.0431_FO1019 | 50 | 1.000 | \left(\delta_{1}, \delta_{2}\right)_{4} | | |
| 0902.0431_FO1020 | 50 | 1.000 | \delta_{1}=\sum_{i}\left[\widetilde{A}_{i}, \widetilde{B}_{i}\right], \delta_{2}=\sum_{j}\left[\widetilde{C}_{j}, \widetilde{D}_{j}\right], A_{i}, B_{i}, C_{j}, D_{j} \in \mathfrak{J}^{C} | | |
| 0902.0431_FO1021 | 50 | 1.000 | \delta_{1}, \delta_{2} | | |
| 0902.0431_FO1022 | 51 | 0.999 | \delta_{1}=[\widetilde{A}, \widetilde{B}], \delta_{2}=[\widetilde{C}, \widetilde{D}], A, B, C, D \in | | |
| 0902.0431_FO1023 | 51 | 0.658 | (X \circ Y, \delta Z)=-(\delta X \circ Y, Z)-(X \circ \delta Y, Z) | | |
| 0902.0431_FO1024 | 51 | 1.000 | B_{4} | | |
| 0902.0431_FO1025 | 51 | 1.000 | k, k^{\prime} \in C | | |
| 0902.0431_FO1026 | 51 | 1.000 | k, k^{\prime} | | |
| 0902.0431_FO1027 | 51 | 1.000 | \delta=\delta_{1}=\delta_{2}=\widetilde{A}_{1}(1) | | |
| 0902.0431_FO1028 | 51 | 1.000 | \widetilde{A}_{1}(1)=-2\left[\widetilde{E}_{3}, \widetilde{F}_{1}(1)\right] | | |
| 0902.0431_FO1029 | 51 | 1.000 | (\operatorname{ad} \delta)^{2} | | |
| 0902.0431_FO1030 | 51 | 1.000 | k=9 | | |
| 0902.0431_FO1031 | 51 | 1.000 | \operatorname{tr}(\delta \delta) | | |
| 0902.0431_FO1032 | 52 | 1.000 | k^{\prime}=3 | | |
| 0902.0431_FO1033 | 52 | 1.000 | A \in \mathfrak{J}_{0}{ }^{C} | | |
| 0902.0431_FO1034 | 52 | 1.000 | B \in \mathfrak{J}_{0}{ }^{C} | | |
| 0902.0431_FO1035 | 52 | 1.000 | [\widetilde{A}, \widetilde{B}] \neq 0 | | |
| 0902.0431_FO1036 | 52 | 1.000 | [\widetilde{A}, \widetilde{B}]=0 | | |
| 0902.0431_FO1037 | 52 | 1.000 | 0=(\delta,[\widetilde{A}, \widetilde{B}])_{4}= | | |
| 0902.0431_FO1038 | 52 | 0.999 | -(\delta,[\widetilde{B}, \widetilde{A}])_{4}=-(\delta B, A) | | |
| 0902.0431_FO1039 | 52 | 1.000 | \left(\mathfrak{J}_{0}{ }^{C}, A\right)=0 | | |
| 0902.0431_FO1040 | 52 | 1.000 | \mathfrak{D}_{4}{ }^{C} | | |
| 0902.0431_FO1041 | 52 | 1.000 | H_{k}=-i G_{k 4+k} | | |
| 0902.0431_FO1042 | 52 | 1.000 | k=0,1,2,3 | | |
| 0902.0431_FO1043 | 53 | 0.963 | \nu, \pi | | |
| 0902.0431_FO1044 | 53 | 0.996 | H_{0}, H_{1}, H_{2}, H_{3} | | |
| 0902.0431_FO1045 | 53 | 1.000 | \mathfrak{h}=\left\{H=\sum_{k=0}^{3} \lambda_{k} H_{k} \mid \lambda_{k} \in C\right\} \subset \mathfrak{d}_{4}{ }^{C} \subset \mathfrak{f}_{4}{ }^{C} | | |
| 0902.0431_FO1046 | 53 | 0.997 | \left[H, \widetilde{A}_{1}(a)\right]=\widetilde{A}_{1}(H a) | | |
| 0902.0431_FO1047 | 53 | 0.996 | \lambda_{k} | | |
| 0902.0431_FO1048 | 53 | 0.996 | \widetilde{A}_{1}\left(e_{k}+i e_{4+k}\right) | | |
| 0902.0431_FO1049 | 53 | 0.996 | 0 \leq k \leq 3 | | |
| 0902.0431_FO1050 | 53 | 0.996 | -\lambda_{k} | | |
| 0902.0431_FO1051 | 53 | 0.996 | \widetilde{A}_{1}\left(e_{k}-i e_{4+k}\right) | | |
| 0902.0431_FO1052 | 54 | 1.000 | \left[H, \widetilde{A}_{2}(a)\right]=\widetilde{A}_{2}((\nu H) a) | | |
| 0902.0431_FO1053 | 54 | 1.000 | \widetilde{A}_{2}\left(e_{k}+\right. | | |
| 0902.0431_FO1054 | 54 | 0.593 | i+i e_{4+k} | | |
| 0902.0431_FO1055 | 54 | 1.000 | \widetilde{A}_{2}\left(e_{k}-i e_{4+k}\right) | | |
| 0902.0431_FO1056 | 54 | 1.000 | \left[H, \widetilde{A}_{3}(a)\right]=\widetilde{A}_{3}((\kappa \pi H) a) | | |
| 0902.0431_FO1057 | 54 | 0.585 | \alpha_{1}, \alpha_{2}, \alpha_{3}, \alpha_{4} | | |
| 0902.0431_FO1058 | 55 | 1.000 | \Pi=\left\{\alpha_{1}, \alpha_{2}, \alpha_{3}, \alpha_{4}\right\} | | |
| 0902.0431_FO1059 | 55 | 1.000 | B_{4}\left(\delta_{1}, \delta_{2}\right)=3 \operatorname{tr}\left(\delta_{1} \delta_{2}\right) | | |
| 0902.0431_FO1060 | 56 | 0.648 | \alpha_{i}\left(B_{4}\left(H_{\alpha}, H\right)=\alpha(H), H \in\right. | | |
| 0902.0431_FO1061 | 56 | 1.000 | -\mu | | |
| 0902.0431_FO1062 | 56 | 1.000 | C_{1} \oplus C_{3} | | |
| 0902.0431_FO1063 | 56 | 1.000 | A_{2} \oplus A_{2} | | |
| 0902.0431_FO1064 | 57 | 0.876 | \left\{\alpha \in F_{4} \mid \alpha E_{i}=E_{i}, i=1,2,3\right\} \cong \operatorname{Spin}(8) | | |
| 0902.0431_FO1065 | 57 | 1.000 | \varphi: \operatorname{Spin}(8) \rightarrow D_{4}=\left\{\alpha \in F_{4} \mid \alpha E_{i}=E_{i}, i=1,2,3\right\} | | |
| 0902.0431_FO1066 | 57 | 1.000 | \alpha=\varphi\left(\alpha_{1}, \alpha_{2}, \alpha_{3}\right) \in D_{4} | | |
| 0902.0431_FO1067 | 57 | 1.000 | \alpha E_{i}=E_{i}, i=1,2,3 | | |
| 0902.0431_FO1068 | 57 | 1.000 | \alpha \in D_{4} | | |
| 0902.0431_FO1069 | 57 | 1.000 | \alpha X \in \mathfrak{J}_{i}, X \in \mathfrak{J}_{i}, \alpha | | |
| 0902.0431_FO1070 | 57 | 1.000 | \alpha: \mathfrak{J}_{i} \rightarrow \mathfrak{J}_{i} | | |
| 0902.0431_FO1071 | 57 | 1.000 | \alpha_{i}: \mathfrak{C} \rightarrow \mathfrak{C} | | |
| 0902.0431_FO1072 | 57 | 1.000 | F_{i}\left(\alpha_{i} x\right) \circ F_{i}\left(\alpha_{i} y\right)=\left(\alpha_{i} x, \alpha_{i} y\right)\left(E_{i+1}+E_{i+2}\right) | | |
| 0902.0431_FO1073 | 57 | 1.000 | \alpha_{i} \in O(8), i=1,2,3 | | |
| 0902.0431_FO1074 | 57 | 1.000 | \varphi\left(\alpha_{1}, \alpha_{2}, \alpha_{3}\right)=\alpha | | |
| 0902.0431_FO1075 | 57 | 1.000 | \operatorname{Ker} \varphi=\{(1,1,1)\} | | |
| 0902.0431_FO1076 | 57 | 1.000 | \left(F_{4}\right)_{E_{1}} | | |
| 0902.0431_FO1077 | 57 | 1.000 | \mathfrak{J}_{01}, \mathfrak{J}_{23} | | |
| 0902.0431_FO1078 | 58 | 1.000 | A \in \mathfrak{J}_{01} | | |
| 0902.0431_FO1079 | 58 | 1.000 | (A, A)= | | |
| 0902.0431_FO1080 | 58 | 1.000 | X_{0} \in \mathfrak{J}_{01} | | |
| 0902.0431_FO1081 | 58 | 1.000 | \left(X_{0}, X_{0}\right)=2,\left(A, X_{0}\right)=0 | | |
| 0902.0431_FO1082 | 58 | 1.000 | Y_{0} \in \mathfrak{J}_{23} | | |
| 0902.0431_FO1083 | 58 | 1.000 | \left(Y_{0}, Y_{0}\right)=2,2 A \circ Y_{0}=-Y_{0} | | |
| 0902.0431_FO1084 | 58 | 1.000 | X_{1} \in \mathfrak{J}_{01} | | |
| 0902.0431_FO1085 | 58 | 1.000 | \left(X_{1}, X_{1}\right)=2,\left(A, X_{1}\right)=\left(X_{0}, X_{1}\right)=0 | | |
| 0902.0431_FO1086 | 58 | 1.000 | X_{2} \in \mathfrak{J}_{01} | | |
| 0902.0431_FO1087 | 58 | 1.000 | \left(X_{2}, X_{2}\right)=2,\left(A, X_{2}\right)=\left(X_{0}, X_{2}\right)= | | |
| 0902.0431_FO1088 | 58 | 1.000 | \left(X_{1}, X_{2}\right)=0 | | |
| 0902.0431_FO1089 | 58 | 1.000 | X_{4} \in \mathfrak{J}_{01} | | |
| 0902.0431_FO1090 | 58 | 1.000 | \left(X_{4}, X_{4}\right)=2,\left(A, X_{4}\right)=\left(X_{0}, X_{4}\right)= | | |
| 0902.0431_FO1091 | 58 | 1.000 | \left(X_{1}, X_{4}\right)=\left(X_{2}, X_{4}\right)=\left(X_{3}, X_{4}\right)=0 | | |
| 0902.0431_FO1092 | 58 | 1.000 | \alpha: \mathfrak{J} \rightarrow \mathfrak{J} | | |
| 0902.0431_FO1093 | 58 | 1.000 | X, Y=E_{i} | | |
| 0902.0431_FO1094 | 58 | 1.000 | F_{j}\left(e_{k}\right) | | |
| 0902.0431_FO1095 | 58 | 1.000 | 27^{2}=729 | | |
| 0902.0431_FO1096 | 58 | 0.995 | \left(F_{4}\right)_{E_{1}} / \operatorname{Spin}(8) \simeq S^{8} | | |
| 0902.0431_FO1097 | 58 | 1.000 | S^{8}=\left\{X \in \mathfrak{J}_{01} \mid(X, X)=2\right\} | | |
| 0902.0431_FO1098 | 58 | 1.000 | \alpha \in\left(F_{4}\right)_{E_{1}} | | |
| 0902.0431_FO1099 | 58 | 1.000 | X \in S^{8} | | |
| 0902.0431_FO1100 | 58 | 1.000 | \alpha X \in S^{8} | | |
| 0902.0431_FO1101 | 58 | 1.000 | S^{8} | | |
| 0902.0431_FO1102 | 58 | 0.999 | A \in S^{8} | | |
| 0902.0431_FO1103 | 59 | 1.000 | \alpha\left(E_{2}-E_{3}\right)=A | | |
| 0902.0431_FO1104 | 59 | 1.000 | E_{2}-E_{3} \in S^{8} | | |
| 0902.0431_FO1105 | 59 | 1.000 | \alpha\left(E_{2}-E_{3}\right)=E_{2}-E_{3} | | |
| 0902.0431_FO1106 | 59 | 1.000 | \alpha E_{1}=E_{1} | | |
| 0902.0431_FO1107 | 59 | 1.000 | \alpha\left(E_{2}+E_{3}\right)=E_{2}+ | | |
| 0902.0431_FO1108 | 59 | 1.000 | E_{3} | | |
| 0902.0431_FO1109 | 59 | 1.000 | \alpha E_{2}=E_{2} | | |
| 0902.0431_FO1110 | 59 | 1.000 | \alpha E_{3}=E_{3} | | |
| 0902.0431_FO1111 | 59 | 1.000 | \alpha \in \operatorname{Spin}(8) | | |
| 0902.0431_FO1112 | 59 | 0.994 | E_{2}-E_{3} | | |
| 0902.0431_FO1113 | 59 | 0.994 | \operatorname{dim}\left(\left(F_{4}\right)_{E_{1}} / \operatorname{Spin}(8)\right)= | | |
| 0902.0431_FO1114 | 59 | 0.892 | \operatorname{dim}\left(F_{4}\right)_{E_{1}}-\operatorname{dim} \operatorname{Spin}(8)=36-28=\operatorname{dim} S^{8} | | |
| 0902.0431_FO1115 | 59 | 0.892 | \left(F_{4}\right)_{E_{1}} / \operatorname{Spin}(8) \simeq | | |
| 0902.0431_FO1116 | 59 | 1.000 | \quad\left(F_{4}\right)_{E_{1}} \cong \operatorname{Spin}(9) | | |
| 0902.0431_FO1117 | 59 | 1.000 | O(9)=O\left(\mathfrak{J}_{01}\right)=\left\{\alpha^{\prime} \in \operatorname{Iso}_{\boldsymbol{R}}\left(\mathfrak{J}_{01}\right) \mid\left(\alpha^{\prime} X, \alpha^{\prime} Y\right)=(X, Y)\right\} | | |
| 0902.0431_FO1118 | 59 | 1.000 | \alpha^{\prime}=\alpha \mid \mathfrak{J}_{01} | | |
| 0902.0431_FO1119 | 59 | 1.000 | \mathfrak{J}_{01} | | |
| 0902.0431_FO1120 | 59 | 1.000 | \alpha^{\prime} \in O(9) | | |
| 0902.0431_FO1121 | 59 | 1.000 | p:\left(F_{4}\right)_{E_{1}} \rightarrow O(9) | | |
| 0902.0431_FO1122 | 59 | 1.000 | p(\alpha)=\alpha^{\prime} | | |
| 0902.0431_FO1123 | 59 | 1.000 | S O(8)=\left\{\alpha^{\prime} \in S O(9) \mid \alpha^{\prime}\left(E_{2}-E_{3}\right)=E_{2}-E_{3}\right\} | | |
| 0902.0431_FO1124 | 59 | 0.643 | p^{\prime} | | |
| 0902.0431_FO1125 | 59 | 0.643 | p:\left(F_{4}\right)_{E_{1}} \rightarrow S O(9) | | |
| 0902.0431_FO1126 | 59 | 0.643 | \operatorname{Spin}(8)=\left\{\alpha \in\left(F_{4}\right)_{E_{1}} \mid \alpha\left(E_{2}-E_{3}\right)=E_{2}-E_{3}\right\} | | |
| 0902.0431_FO1127 | 59 | 0.999 | p: \operatorname{Spin}(9) \rightarrow S O(8) | | |
| 0902.0431_FO1128 | 59 | 0.912 | p^{\prime}: \operatorname{Spin}(8) \rightarrow S O(8) | | |
| 0902.0431_FO1129 | 59 | 1.000 | \operatorname{Ker} p=\{1, \sigma\} | | |
| 0902.0431_FO1130 | 59 | 1.000 | \sigma= | | |
| 0902.0431_FO1131 | 59 | 1.000 | \varphi(1,-1,-1) | | |
| 0902.0431_FO1132 | 59 | 1.000 | \alpha \in \operatorname{Ker} p | | |
| 0902.0431_FO1133 | 59 | 1.000 | \alpha X=X | | |
| 0902.0431_FO1134 | 59 | 1.000 | X \in \mathfrak{J}_{01} | | |
| 0902.0431_FO1135 | 59 | 1.000 | \alpha=\left(\alpha_{1}, \alpha_{2}, \alpha_{3}\right) \in \operatorname{Spin}(8) | | |
| 0902.0431_FO1136 | 59 | 1.000 | \alpha F_{1}(x)=F_{1}(x) | | |
| 0902.0431_FO1137 | 59 | 1.000 | F_{1}\left(\alpha_{1} x\right)=F_{1}(x) | | |
| 0902.0431_FO1138 | 59 | 1.000 | \alpha_{1} x=x | | |
| 0902.0431_FO1139 | 59 | 1.000 | \alpha_{1}=1 | | |
| 0902.0431_FO1140 | 59 | 1.000 | \alpha=(1,1,1)=1 | | |
| 0902.0431_FO1141 | 59 | 1.000 | \alpha=(1,-1,-1)=\sigma | | |
| 0902.0431_FO1142 | 59 | 1.000 | S O(9) | | |
| 0902.0431_FO1143 | 60 | 0.970 | \quad \operatorname{Spin}(9) / \operatorname{Spin}^{\prime}(7) \simeq S^{15} | | |
| 0902.0431_FO1144 | 60 | 0.985 | \operatorname{Spin}^{\prime}(7)=\{\widetilde{\alpha} \in \operatorname{SO}(8) \mid(\widetilde{\alpha} x)(\alpha y)=\widetilde{\alpha}(x y), x, y \in \mathfrak{C} | | |
| 0902.0431_FO1145 | 60 | 0.985 | \alpha \in S O(7)\} | | |
| 0902.0431_FO1146 | 60 | 0.999 | S^{15}=\left\{Y \in \mathfrak{J}_{23} \mid(Y, Y)=2\right\} | | |
| 0902.0431_FO1147 | 60 | 0.999 | \alpha \in \operatorname{Spin}(9) | | |
| 0902.0431_FO1148 | 60 | 0.999 | Y \in S^{15} | | |
| 0902.0431_FO1149 | 60 | 1.000 | \alpha Y \in S^{15} | | |
| 0902.0431_FO1150 | 60 | 1.000 | S^{15} | | |
| 0902.0431_FO1151 | 60 | 1.000 | Y_{0} \in S^{15} | | |
| 0902.0431_FO1152 | 60 | 1.000 | (A, A)=2,2 A \circ Y_{0}=-Y_{0} | | |
| 0902.0431_FO1153 | 60 | 1.000 | Y_{0} | | |
| 0902.0431_FO1154 | 60 | 1.000 | X_{i}, Y_{i}, Z_{i} | | |
| 0902.0431_FO1155 | 60 | 1.000 | \alpha F_{2}(1)=Y_{0} | | |
| 0902.0431_FO1156 | 60 | 0.999 | \operatorname{Spin}(9)_{F_{2}(1)} | | |
| 0902.0431_FO1157 | 60 | 0.999 | F_{2}(1) \in S^{15} | | |
| 0902.0431_FO1158 | 60 | 0.999 | \alpha F_{2}(1)= | | |
| 0902.0431_FO1159 | 60 | 1.000 | F_{2}(1) | | |
| 0902.0431_FO1160 | 60 | 1.000 | F_{2}(1) \circ F_{2}(1)=E_{1}+E_{3} | | |
| 0902.0431_FO1161 | 60 | 1.000 | F_{2}(1) \circ F_{2}(1)=E_{1}+\alpha E_{3} | | |
| 0902.0431_FO1162 | 60 | 1.000 | \alpha\left(E_{2}+E_{3}\right)=E_{2}+E_{3} | | |
| 0902.0431_FO1163 | 60 | 0.999 | \alpha=\left(\alpha_{1}, \alpha_{2}, \alpha_{3}\right) | | |
| 0902.0431_FO1164 | 60 | 0.999 | F_{2}\left(\alpha_{2}(1)\right)=F_{2}(1) | | |
| 0902.0431_FO1165 | 60 | 0.999 | \alpha_{2} 1=1 | | |
| 0902.0431_FO1166 | 60 | 0.943 | \alpha_{2} \in S O(7) | | |
| 0902.0431_FO1167 | 60 | 0.943 | \alpha=\left(\alpha_{1}, \alpha_{2}, \alpha_{3}\right) \in \operatorname{Spin}^{\prime}(7) | | |
| 0902.0431_FO1168 | 60 | 0.943 | \alpha \in \operatorname{Spin}(7) | | |
| 0902.0431_FO1169 | 60 | 1.000 | \alpha F_{2}(1)=F_{2}(1) | | |
| 0902.0431_FO1170 | 60 | 1.000 | \operatorname{Spin}(9)_{F_{2}(1)}=\operatorname{Spin}^{\prime}(7) | | |
| 0902.0431_FO1171 | 60 | 0.991 | \operatorname{Spin}(9) / \operatorname{Spin}^{\prime}(7) \simeq S^{15} | | |
| 0902.0431_FO1172 | 60 | 0.870 | \left(\operatorname{Spin}^{\prime}(7)\right. | | |
| 0902.0431_FO1173 | 60 | 0.870 | f: | | |
| 0902.0431_FO1174 | 60 | 0.992 | \operatorname{Spin}^{\prime}(7) \rightarrow \operatorname{Spin}(7) | | |
| 0902.0431_FO1175 | 60 | 0.687 | \left.\operatorname{Spin}^{\prime}(7) \cong \operatorname{Spin}(7)\right) | | |
| 0902.0431_FO1176 | 60 | 0.856 | \left(F_{4}\right)_{0} | | |
| 0902.0431_FO1177 | 60 | 1.000 | \beta_{1}(a): \mathfrak{J} \rightarrow \mathfrak{J} | | |
| 0902.0431_FO1178 | 60 | 1.000 | \beta_{1}(a) X(\xi, x)= | | |
| 0902.0431_FO1179 | 60 | 1.000 | Y(\eta, y) | | |
| 0902.0431_FO1180 | 60 | 0.995 | a=0 | | |
| 0902.0431_FO1181 | 60 | 0.995 | \frac{\sin |a|}{|a|} | | |
| 0902.0431_FO1182 | 60 | 0.995 | \beta_{1}(a) \in\left(F_{4}\right)_{0} | | |
| 0902.0431_FO1183 | 60 | 1.000 | A_{1}(a)=\left(\begin{array}{ccc}0 & 0 & 0 \\ 0 & 0 & a \\ 0 & -\bar{a} & 0\end{array}\right) | | |
| 0902.0431_FO1184 | 60 | 1.000 | \widetilde{A}_{1}(a) \in \mathfrak{f}_{4} | | |
| 0902.0431_FO1185 | 61 | 0.896 | \beta_{1}(a)=\exp \widetilde{A}_{1}(a) | | |
| 0902.0431_FO1186 | 61 | 1.000 | \alpha \in\left(F_{4}\right)_{0} | | |
| 0902.0431_FO1187 | 61 | 1.000 | \xi_{1}, \xi_{2}, \xi_{3} | | |
| 0902.0431_FO1188 | 61 | 1.000 | \mathfrak{X}=\left\{\alpha X \mid \alpha \in\left(F_{4}\right)_{0}\right\} | | |
| 0902.0431_FO1189 | 61 | 1.000 | \mathfrak{X} | | |
| 0902.0431_FO1190 | 61 | 1.000 | \xi_{1}{ }^{2}+\xi_{2}{ }^{2}+\xi_{3}{ }^{2} | | |
| 0902.0431_FO1191 | 61 | 0.999 | \eta_{1}{ }^{2}+\eta_{2}{ }^{2}+\eta_{3}{ }^{2} | | |
| 0902.0431_FO1192 | 61 | 0.999 | Y=Y(\eta, y) \in \mathfrak{X} | | |
| 0902.0431_FO1193 | 61 | 0.999 | X_{0}=X(\xi, x) | | |
| 0902.0431_FO1194 | 61 | 1.000 | X_{0} | | |
| 0902.0431_FO1195 | 61 | 0.999 | 2 \times 3 | | |
| 0902.0431_FO1196 | 61 | 0.999 | x_{1} | | |
| 0902.0431_FO1197 | 61 | 1.000 | a(t)=\frac{x_{1}}{\left|x_{1}\right|} t, t>0 | | |
| 0902.0431_FO1198 | 61 | 1.000 | \beta_{1}(a(t)) \in\left(F_{4}\right)_{0} | | |
| 0902.0431_FO1199 | 61 | 1.000 | |a(t)|=t | | |
| 0902.0431_FO1200 | 61 | 1.000 | \frac{\left(a(t), x_{1}\right)}{|a(t)|}=\left|x_{1}\right| | | |
| 0902.0431_FO1201 | 61 | 1.000 | Y(\eta(t), y(t))=\beta_{1}(a(t)) X_{0} \in \mathfrak{X} | | |
| 0902.0431_FO1202 | 61 | 1.000 | t>0 | | |
| 0902.0431_FO1203 | 61 | 1.000 | x_{1}=0 . x_{2}=x_{3}=0 | | |
| 0902.0431_FO1204 | 61 | 0.999 | \beta_{2}(a), \beta_{3}(a) \in\left(F_{4}\right)_{0} | | |
| 0902.0431_FO1205 | 61 | 0.999 | \beta_{1}(a) | | |
| 0902.0431_FO1206 | 61 | 1.000 | \alpha X=\left(\begin{array}{ccc}\xi_{1} & 0 & 0 \\ 0 & \xi_{2} & 0 \\ 0 & 0 & \xi_{3}\end{array}\right) | | |
| 0902.0431_FO1207 | 62 | 1.000 | \mathfrak{C} P_{2} | | |
| 0902.0431_FO1208 | 62 | 1.000 | X \in \mathfrak{C} P_{2} | | |
| 0902.0431_FO1209 | 62 | 1.000 | \alpha X \in \mathfrak{C} P_{2} | | |
| 0902.0431_FO1210 | 62 | 1.000 | E_{1} \in \mathfrak{C} P_{2} | | |
| 0902.0431_FO1211 | 62 | 1.000 | X \in \mathfrak{C} P_{2} \subset \mathfrak{J} | | |
| 0902.0431_FO1212 | 62 | 1.000 | X \circ X=X | | |
| 0902.0431_FO1213 | 62 | 1.000 | \alpha X \circ \alpha X=\alpha X | | |
| 0902.0431_FO1214 | 62 | 1.000 | \xi_{i}{ }^{2}=\xi_{i} | | |
| 0902.0431_FO1215 | 62 | 1.000 | \xi_{i}=1 | | |
| 0902.0431_FO1216 | 62 | 1.000 | \xi_{i}=0, i=1,2,3 | | |
| 0902.0431_FO1217 | 62 | 1.000 | \operatorname{tr}(\alpha X)=\operatorname{tr}(X)=1 | | |
| 0902.0431_FO1218 | 62 | 1.000 | \xi_{i}=1, \xi_{i+1}=\xi_{i+2}=0 | | |
| 0902.0431_FO1219 | 62 | 1.000 | i | | |
| 0902.0431_FO1220 | 62 | 1.000 | \alpha X=E_{i} | | |
| 0902.0431_FO1221 | 62 | 1.000 | E_{2} | | |
| 0902.0431_FO1222 | 62 | 1.000 | E_{1} | | |
| 0902.0431_FO1223 | 62 | 0.997 | \beta: \mathfrak{J} \rightarrow \mathfrak{J}, \beta X=T X T^{-1} | | |
| 0902.0431_FO1224 | 62 | 0.997 | T=\left(\begin{array}{ccc}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & -1\end{array}\right) \in S O(3) | | |
| 0902.0431_FO1225 | 62 | 0.997 | \beta \in\left(F_{4}\right)_{0} | | |
| 0902.0431_FO1226 | 62 | 1.000 | \beta E_{2}=E_{1} | | |
| 0902.0431_FO1227 | 62 | 1.000 | \beta \alpha X=E_{1} | | |
| 0902.0431_FO1228 | 62 | 1.000 | \alpha X=E_{3} | | |
| 0902.0431_FO1229 | 62 | 1.000 | \mathfrak{C} P_{2}=\left(F_{4}\right)_{0} E_{1}, \mathfrak{C} P_{2} | | |
| 0902.0431_FO1230 | 62 | 0.999 | F_{4} / \operatorname{Spin}(9) \simeq \mathfrak{C} P_{2} | | |
| 0902.0431_FO1231 | 63 | 1.000 | \sigma \in F_{4} | | |
| 0902.0431_FO1232 | 63 | 1.000 | \sigma^{2}=1 | | |
| 0902.0431_FO1233 | 63 | 1.000 | \sigma=(1,-1,-1) \in | | |
| 0902.0431_FO1234 | 63 | 1.000 | \operatorname{Spin}(8) \subset \operatorname{Spin}(9) \subset F_{4} | | |
| 0902.0431_FO1235 | 63 | 1.000 | \left(F_{4}\right)^{\sigma} | | |
| 0902.0431_FO1236 | 63 | 0.991 | \mathfrak{J}_{\sigma} | | |
| 0902.0431_FO1237 | 63 | 0.991 | \mathfrak{J}_{-\sigma} | | |
| 0902.0431_FO1238 | 63 | 1.000 | \mathfrak{J}=\mathfrak{J}_{\sigma} \oplus \mathfrak{J}_{-\sigma} | | |
| 0902.0431_FO1239 | 63 | 1.000 | \mathfrak{J}_{\sigma}, \mathfrak{J}_{-\sigma} | | |
| 0902.0431_FO1240 | 63 | 0.992 | \quad\left(F_{4}\right)^{\sigma}=\left(F_{4}\right)_{E_{1}}=\operatorname{Spin}(9) | | |
| 0902.0431_FO1241 | 63 | 1.000 | \alpha \in\left(F_{4}\right)^{\sigma} | | |
| 0902.0431_FO1242 | 63 | 1.000 | \alpha E_{3} \in \mathfrak{J}(2, \mathfrak{C}) | | |
| 0902.0431_FO1243 | 63 | 1.000 | \alpha E_{1}=\alpha\left(E-E_{2}-E_{3}\right)=E-\alpha E_{2}-\alpha E_{3} | | |
| 0902.0431_FO1244 | 64 | 1.000 | \xi_{2}=\xi_{3}=x_{1}=0 | | |
| 0902.0431_FO1245 | 64 | 1.000 | \mathfrak{J}_{\sigma}=\left\{X \in \mathfrak{J} \mid E_{1} \circ X=0\right\} \oplus \mathfrak{E}_{1}, \mathfrak{J}_{-\sigma}=\left\{X \in \mathfrak{J} \mid 2 E_{1} \circ X=X\right\} | | |
| 0902.0431_FO1246 | 64 | 1.000 | \alpha \sigma=\sigma \alpha | | |
| 0902.0431_FO1247 | 64 | 1.000 | \left(F_{4}\right)^{\sigma}=\left(F_{4}\right)_{E_{1}} \cong | | |
| 0902.0431_FO1248 | 64 | 1.000 | \alpha \in z\left(F_{4}\right) | | |
| 0902.0431_FO1249 | 64 | 1.000 | \sigma: \sigma \alpha=\alpha \sigma | | |
| 0902.0431_FO1250 | 64 | 1.000 | \alpha \in z(\operatorname{Spin}(9)) | | |
| 0902.0431_FO1251 | 64 | 1.000 | z(\operatorname{Spin}(9)) | | |
| 0902.0431_FO1252 | 64 | 1.000 | \sigma \notin z\left(F_{4}\right) | | |
| 0902.0431_FO1253 | 64 | 1.000 | F_{4}=\left\{\alpha \in \operatorname{Iso}_{\boldsymbol{R}}(\mathfrak{J}) \mid \alpha(X \circ Y)=\alpha X \circ \alpha Y\right\} | | |
| 0902.0431_FO1254 | 64 | 1.000 | \gamma \in G_{2} \subset F_{4} | | |
| 0902.0431_FO1255 | 65 | 1.000 | \left(F_{4}\right)^{\gamma} | | |
| 0902.0431_FO1256 | 65 | 1.000 | x_{i}=m_{i}+a_{i} e_{4} \in \boldsymbol{H} \oplus \boldsymbol{H} e_{4}=\mathfrak{C} | | |
| 0902.0431_FO1257 | 65 | 1.000 | \mathfrak{J}(3, \boldsymbol{H}) \oplus \boldsymbol{H}^{3} | | |
| 0902.0431_FO1258 | 65 | 0.998 | \mathfrak{J}(3, \boldsymbol{H}) | | |
| 0902.0431_FO1259 | 65 | 0.998 | M \times N | | |
| 0902.0431_FO1260 | 65 | 0.998 | (M, N) | | |
| 0902.0431_FO1261 | 65 | 1.000 | \boldsymbol{H}^{3} | | |
| 0902.0431_FO1262 | 65 | 1.000 | \frac{1}{2}\left(\boldsymbol{a}^{*} \boldsymbol{b}+\right. | | |
| 0902.0431_FO1263 | 65 | 0.813 | \left.\left.\boldsymbol{b}^{*} \boldsymbol{a}\right)\right) | | |
| 0902.0431_FO1264 | 65 | 1.000 | F_{4, \boldsymbol{H}} | | |
| 0902.0431_FO1265 | 65 | 0.568 | \mathfrak{J}_{\boldsymbol{H}}=\mathfrak{J}(3, \boldsymbol{H}) | | |
| 0902.0431_FO1266 | 65 | 1.000 | \quad F_{4, \boldsymbol{H}} \cong \operatorname{Sp}(3) / \boldsymbol{Z}_{2}, \quad \boldsymbol{Z}_{2}=\{E,-E\} | | |
| 0902.0431_FO1267 | 65 | 0.964 | \varphi: \operatorname{Sp}(3) \rightarrow F_{4, \boldsymbol{H}} | | |
| 0902.0431_FO1268 | 65 | 1.000 | \alpha \in F_{4, \boldsymbol{H}} | | |
| 0902.0431_FO1269 | 65 | 1.000 | \alpha E_{i}, i=1,2,3 | | |
| 0902.0431_FO1270 | 65 | 1.000 | \alpha E_{i} | | |
| 0902.0431_FO1271 | 65 | 1.000 | \alpha E_{i} \circ \alpha E_{i}=\alpha E_{i} | | |
| 0902.0431_FO1272 | 65 | 1.000 | \operatorname{tr}\left(\alpha E_{i}\right)=1 | | |
| 0902.0431_FO1273 | 65 | 1.000 | \alpha E_{i} \in \boldsymbol{H} P_{2}=\left\{M \in \mathfrak{J}_{\boldsymbol{H}} \mid M \circ M=\right. | | |
| 0902.0431_FO1274 | 66 | 0.852 | M, \operatorname{tr}(M)=1\} | | |
| 0902.0431_FO1275 | 66 | 0.852 | A_{i} \in \operatorname{Sp}(3) | | |
| 0902.0431_FO1276 | 66 | 0.999 | \alpha E_{i}=A_{i} E_{i} A_{i}{ }^{*} | | |
| 0902.0431_FO1277 | 66 | 0.999 | \boldsymbol{a}_{i}=\left(\begin{array}{c}a_{i 1} \\ a_{i 2} \\ a_{i 3}\end{array}\right) | | |
| 0902.0431_FO1278 | 66 | 0.999 | A_{i} | | |
| 0902.0431_FO1279 | 66 | 0.999 | \alpha E_{i}=\left(\begin{array}{ccc}a_{i 1} \bar{a}_{i 1} & a_{i 1} \bar{a}_{i 2} & a_{i 1} \bar{a}_{i 3} \\ a_{i 2} \bar{a}_{i 1} & a_{i 2} \bar{a}_{i 2} & a_{i 2} \bar{a}_{i 3} \\ a_{i 3} \bar{a}_{i 1} & a_{i 3} \bar{a}_{i 2} & a_{i 3} \bar{a}_{i 3}\end{array}\right) | | |
| 0902.0431_FO1280 | 66 | 0.999 | A=\left(\boldsymbol{a}_{1}, \boldsymbol{a}_{2}, \boldsymbol{a}_{3}\right) | | |
| 0902.0431_FO1281 | 66 | 1.000 | A A^{*}=A\left(E_{1}+E_{2}+E_{3}\right) A^{*}=\alpha E_{1}+\alpha E_{2}+\alpha E_{3}=\alpha E=E | | |
| 0902.0431_FO1282 | 66 | 1.000 | A \in \operatorname{Sp}(3) | | |
| 0902.0431_FO1283 | 66 | 1.000 | \beta=\varphi(A)^{-1} \alpha | | |
| 0902.0431_FO1284 | 66 | 1.000 | \beta \in F_{4, \boldsymbol{H}} | | |
| 0902.0431_FO1285 | 66 | 1.000 | \beta_{1}, \beta_{2}, \beta_{3} | | |
| 0902.0431_FO1286 | 66 | 0.999 | : \boldsymbol{H} \rightarrow \boldsymbol{H} | | |
| 0902.0431_FO1287 | 66 | 0.995 | p=\beta_{1} 1 | | |
| 0902.0431_FO1288 | 66 | 0.995 | q=\beta_{2} 1 | | |
| 0902.0431_FO1289 | 66 | 0.995 | |p|=|q|=1 | | |
| 0902.0431_FO1290 | 66 | 0.995 | m=1 | | |
| 0902.0431_FO1291 | 66 | 0.995 | n=1 | | |
| 0902.0431_FO1292 | 66 | 1.000 | p\left(\beta_{2} n\right)=\overline{\beta_{3} \bar{n}} | | |
| 0902.0431_FO1293 | 66 | 1.000 | \left(\beta_{1} m\right) q=\overline{\beta_{3} \bar{m}} | | |
| 0902.0431_FO1294 | 66 | 1.000 | \zeta m=\bar{p}\left(\beta_{1} m\right) | | |
| 0902.0431_FO1295 | 66 | 1.000 | \zeta | | |
| 0902.0431_FO1296 | 66 | 1.000 | \boldsymbol{H}: \zeta \in \operatorname{Aut}(\boldsymbol{H}) | | |
| 0902.0431_FO1297 | 66 | 1.000 | \zeta m=r m \bar{r} | | |
| 0902.0431_FO1298 | 66 | 0.996 | r \in \operatorname{Sp}(1) | | |
| 0902.0431_FO1299 | 66 | 0.999 | B=\left(\begin{array}{ccc}\bar{q} r & 0 & 0 \\ 0 & p r & 0 \\ 0 & 0 & r\end{array}\right) | | |
| 0902.0431_FO1300 | 66 | 0.999 | B \in \operatorname{Sp}(3) | | |
| 0902.0431_FO1301 | 66 | 1.000 | \beta=\varphi(B) | | |
| 0902.0431_FO1302 | 66 | 1.000 | \operatorname{Ker} \varphi=\{E,-E\} | | |
| 0902.0431_FO1303 | 66 | 0.997 | \operatorname{Sp}(3) / \boldsymbol{Z}_{2} \cong F_{4, \boldsymbol{H}} | | |
| 0902.0431_FO1304 | 66 | 0.301 | \left(F_{4}\right)^{\gamma} \cong(\operatorname{Sp}(1) \times \operatorname{Sp}(3)) / \boldsymbol{Z}_{2}, \boldsymbol{Z}_{2}=\{(1, E),(-1,-E)\} | | |
| 0902.0431_FO1305 | 67 | 0.978 | \varphi: \operatorname{Sp}(1) \times \operatorname{Sp}(3) \rightarrow\left(F_{4}\right)^{\gamma} | | |
| 0902.0431_FO1306 | 67 | 0.867 | \varphi(p, A) \in\left(F_{4}\right)^{\gamma} | | |
| 0902.0431_FO1307 | 67 | 0.867 | p \in S p(1), A \in S p(3), M, N \in \mathfrak{J}_{\boldsymbol{H}} | | |
| 0902.0431_FO1308 | 67 | 1.000 | \boldsymbol{a}, \boldsymbol{b} \in \boldsymbol{H}^{3} | | |
| 0902.0431_FO1309 | 67 | 1.000 | \varphi(p, A) | | |
| 0902.0431_FO1310 | 67 | 1.000 | \varphi(p, A) \in F_{4} | | |
| 0902.0431_FO1311 | 67 | 1.000 | \gamma \varphi(p, A)=\varphi(p, A) \gamma | | |
| 0902.0431_FO1312 | 67 | 1.000 | \alpha \in\left(F_{4}\right)^{\gamma} | | |
| 0902.0431_FO1313 | 67 | 1.000 | \alpha^{\prime}=\alpha \mid \mathfrak{J}_{\boldsymbol{H}} | | |
| 0902.0431_FO1314 | 67 | 1.000 | \mathfrak{J}_{\boldsymbol{H}}=\{X \in \mathfrak{J} \mid \gamma X=X\} | | |
| 0902.0431_FO1315 | 67 | 0.975 | \beta=\varphi(1, A)^{-1} \alpha | | |
| 0902.0431_FO1316 | 67 | 0.975 | \beta \mid \mathfrak{J}_{\boldsymbol{H}}=1 | | |
| 0902.0431_FO1317 | 67 | 0.975 | \beta \in G_{2} | | |
| 0902.0431_FO1318 | 67 | 1.000 | \beta \in \operatorname{Spin}(7) | | |
| 0902.0431_FO1319 | 67 | 1.000 | \beta=\left(\beta_{1}, \beta, \kappa \beta\right) | | |
| 0902.0431_FO1320 | 67 | 1.000 | \left(\beta_{1} x\right)(\beta y)= | | |
| 0902.0431_FO1321 | 67 | 1.000 | \beta(x y) | | |
| 0902.0431_FO1322 | 67 | 1.000 | \beta_{1}=\beta | | |
| 0902.0431_FO1323 | 67 | 1.000 | \beta=\varphi(p, E) | | |
| 0902.0431_FO1324 | 67 | 1.000 | \alpha=\varphi(1, A) \beta=\varphi(1, A) \varphi(p, E)=\varphi(p, A) | | |
| 0902.0431_FO1325 | 67 | 0.999 | \operatorname{Ker} \varphi=\{(1, E),(-1,-E)\}=\boldsymbol{Z}_{2} | | |
| 0902.0431_FO1326 | 67 | 0.740 | (S p(1) \times S p(3)) / \boldsymbol{Z}_{2} \cong\left(F_{4}\right)^{\gamma} | | |
| 0902.0431_FO1327 | 67 | 1.000 | \varphi: \operatorname{Sp}(1) \times \operatorname{Sp}(3) \rightarrow F_{4} | | |
| 0902.0431_FO1328 | 67 | 1.000 | \left(\mathfrak{f}_{4}\right)^{\gamma} | | |
| 0902.0431_FO1329 | 67 | 1.000 | \operatorname{dim}\left(\left(\mathfrak{f}_{4}\right)^{\gamma}\right)=6 \times 2+4 \times 3=24=3+21=\operatorname{dim}(\mathfrak{s p}(1) \oplus | | |
| 0902.0431_FO1330 | 67 | 0.508 | \mathfrak{s} \mathfrak{p}(3)) | | |
| 0902.0431_FO1331 | 68 | 1.000 | w \in G_{2} \subset F_{4} | | |
| 0902.0431_FO1332 | 68 | 1.000 | \left(F_{4}\right)^{w} | | |
| 0902.0431_FO1333 | 68 | 1.000 | \mathfrak{J}(3, \boldsymbol{C}) \oplus M(3, \boldsymbol{C}) | | |
| 0902.0431_FO1334 | 68 | 1.000 | M(3, \boldsymbol{C}) | | |
| 0902.0431_FO1335 | 68 | 1.000 | M=\left(\boldsymbol{m}_{1}, \boldsymbol{m}_{2}, \boldsymbol{m}_{3}\right), N=\left(\boldsymbol{n}_{1}, \boldsymbol{n}_{2}, \boldsymbol{n}_{3}\right) \in M(3, \boldsymbol{C}) | | |
| 0902.0431_FO1336 | 68 | 1.000 | P, A \in M(3, \boldsymbol{C}) | | |
| 0902.0431_FO1337 | 68 | 1.000 | M, N \in M(3, \boldsymbol{C}) | | |
| 0902.0431_FO1338 | 68 | 1.000 | \widetilde{P} | | |
| 0902.0431_FO1339 | 68 | 1.000 | P | | |
| 0902.0431_FO1340 | 68 | 0.998 | M=\left(m_{i j}\right), N=\left(n_{i j}\right) \in M(3, \boldsymbol{C}) | | |
| 0902.0431_FO1341 | 69 | 0.999 | \mathfrak{J}(3, \boldsymbol{C}) | | |
| 0902.0431_FO1342 | 69 | 1.000 | F_{4, \boldsymbol{C}} | | |
| 0902.0431_FO1343 | 69 | 0.881 | \mathfrak{J}_{\boldsymbol{C}}=\mathfrak{J}(3, \boldsymbol{C}): | | |
| 0902.0431_FO1344 | 69 | 0.856 | \boldsymbol{Z}_{2}=\{1, \epsilon\} | | |
| 0902.0431_FO1345 | 69 | 0.983 | S U(3) \cdot \boldsymbol{Z}_{2} | | |
| 0902.0431_FO1346 | 69 | 0.983 | \boldsymbol{Z}_{2} | | |
| 0902.0431_FO1347 | 69 | 0.642 | F_{4, \boldsymbol{C}} \cong\left(S U(3) / \boldsymbol{Z}_{3}\right) \cdot \boldsymbol{Z}_{2}, \quad \boldsymbol{Z}_{3}=\left\{E, \omega_{1} E, \omega_{1}{ }^{2} E\right\}, \omega_{1}= | | |
| 0902.0431_FO1348 | 69 | 0.996 | -\frac{1}{2}+\frac{\sqrt{3}}{2} e_{1} | | |
| 0902.0431_FO1349 | 69 | 1.000 | \varphi: S U(3) \cdot \boldsymbol{Z}_{2} \rightarrow F_{4, \boldsymbol{C}} | | |
| 0902.0431_FO1350 | 69 | 0.968 | C, S U(3) | | |
| 0902.0431_FO1351 | 69 | 0.947 | \boldsymbol{H}, \operatorname{Sp}(3)) | | |
| 0902.0431_FO1352 | 69 | 0.947 | \zeta \in \operatorname{Aut}(\boldsymbol{C}) | | |
| 0902.0431_FO1353 | 69 | 1.000 | \zeta=1 | | |
| 0902.0431_FO1354 | 69 | 1.000 | \zeta=\epsilon | | |
| 0902.0431_FO1355 | 69 | 1.000 | \epsilon x=\bar{x}, x \in \boldsymbol{C} | | |
| 0902.0431_FO1356 | 69 | 1.000 | r \in \boldsymbol{C} | | |
| 0902.0431_FO1357 | 69 | 1.000 | r^{-3}=\bar{q} p | | |
| 0902.0431_FO1358 | 69 | 1.000 | B \in S U(3) | | |
| 0902.0431_FO1359 | 69 | 0.817 | \left(F_{4}\right)^{w} \cong(S U(3) \times S U(3)) / \boldsymbol{Z}_{3}, \quad \boldsymbol{Z}_{3}=\left\{(E, E),\left(\omega_{1} E, \omega_{1} E\right)\right. | | |
| 0902.0431_FO1360 | 69 | 0.674 | \left.\left(\omega_{1}{ }^{2} E, \omega_{1}{ }^{2} E\right)\right\}, \omega_{1}=-\frac{1}{2}+\frac{\sqrt{3}}{2} e_{1} | | |
| 0902.0431_FO1361 | 69 | 0.817 | \varphi: S U(3) \times S U(3) \rightarrow\left(F_{4}\right)^{w} | | |
| 0902.0431_FO1362 | 70 | 1.000 | \varphi(P, A) \in\left(F_{4}\right)^{w} | | |
| 0902.0431_FO1363 | 70 | 1.000 | P, A \in S U(3) | | |
| 0902.0431_FO1364 | 70 | 1.000 | X+M, Y+N \in | | |
| 0902.0431_FO1365 | 70 | 1.000 | \varphi(P, A) | | |
| 0902.0431_FO1366 | 70 | 1.000 | \varphi(P, A) \in F_{4} | | |
| 0902.0431_FO1367 | 70 | 1.000 | w \varphi(P, A)=\varphi(P, A) w | | |
| 0902.0431_FO1368 | 70 | 1.000 | \alpha \in\left(F_{4}\right)^{w} | | |
| 0902.0431_FO1369 | 70 | 1.000 | \alpha^{\prime}=\alpha \mid \mathfrak{J}_{\boldsymbol{C}} | | |
| 0902.0431_FO1370 | 70 | 1.000 | \mathfrak{J}_{\boldsymbol{C}}=\{X \in \mathfrak{J} \mid w X=X\} | | |
| 0902.0431_FO1371 | 70 | 0.603 | \beta=\varphi(E, A)^{-1} \alpha | | |
| 0902.0431_FO1372 | 70 | 0.603 | \beta \mid \mathfrak{J}_{C}=1 | | |
| 0902.0431_FO1373 | 70 | 1.000 | \beta \in\left(G_{2}\right)_{e_{1}}=\left(G_{2}\right)^{w}=S U(3) | | |
| 0902.0431_FO1374 | 70 | 1.000 | P \in S U(3) | | |
| 0902.0431_FO1375 | 70 | 1.000 | \gamma_{1}: \mathfrak{J} \rightarrow \mathfrak{J} | | |
| 0902.0431_FO1376 | 70 | 1.000 | \gamma_{1}(X+M)=\bar{X}+\bar{M} | | |
| 0902.0431_FO1377 | 70 | 1.000 | X+M \in \mathfrak{J} | | |
| 0902.0431_FO1378 | 70 | 1.000 | \gamma_{1} \in G_{2} \subset F_{4} | | |
| 0902.0431_FO1379 | 70 | 1.000 | \beta=\alpha^{-1} \varphi(E, A) \gamma_{1} | | |
| 0902.0431_FO1380 | 70 | 1.000 | \beta \in F_{4} | | |
| 0902.0431_FO1381 | 70 | 0.662 | \beta \mid \widetilde{\mathfrak{J}}_{\boldsymbol{C}}=1 | | |
| 0902.0431_FO1382 | 70 | 0.662 | \beta \in\left(G_{2}\right)_{e_{1}}=\left(G_{2}\right)^{w} | | |
| 0902.0431_FO1383 | 70 | 0.662 | \subset\left(F_{4}\right)^{w} | | |
| 0902.0431_FO1384 | 70 | 1.000 | \alpha, \varphi(E, A) \in\left(F_{4}\right)^{w} | | |
| 0902.0431_FO1385 | 70 | 1.000 | \gamma_{1} \in\left(F_{4}\right)^{w} | | |
| 0902.0431_FO1386 | 70 | 1.000 | \gamma_{1} \in\left(G_{2}\right)^{w} | | |
| 0902.0431_FO1387 | 70 | 0.997 | \operatorname{Ker} \varphi=\left\{(E, E),\left(\omega_{1} E, \omega_{1} E\right),\left(\omega_{1}{ }^{2} E, \omega_{1}{ }^{2} E\right)\right\}=\boldsymbol{Z}_{3} | | |
| 0902.0431_FO1388 | 70 | 1.000 | (S U(3) \times S U(3)) / \boldsymbol{Z}_{3} \cong\left(F_{4}\right)^{w} | | |
| 0902.0431_FO1389 | 71 | 1.000 | \left(\mathfrak{f}_{4}\right)^{w} | | |
| 0902.0431_FO1390 | 71 | 1.000 | \operatorname{dim}\left(\mathfrak{f}_{4}\right)^{w}=10+6=16=8+8=\operatorname{dim}(\mathfrak{s u}(3)+\mathfrak{s u}(3)) | | |
| 0902.0431_FO1391 | 71 | 0.977 | \left.(S U(3) \times S U(3)) / \boldsymbol{Z}_{3}\right) \cdot \boldsymbol{Z}_{2} | | |
| 0902.0431_FO1392 | 71 | 0.977 | (S U(3) \times | | |
| 0902.0431_FO1393 | 71 | 0.993 | \boldsymbol{Z}_{2}=\left\{1, \gamma_{1}\right\} | | |
| 0902.0431_FO1394 | 71 | 0.993 | (S U(3) \times S U(3)) | | |
| 0902.0431_FO1395 | 71 | 1.000 | \left.\gamma_{1}(P, A)=(\bar{P}, \bar{A})\right) | | |
| 0902.0431_FO1396 | 71 | 1.000 | \langle X, Y\rangle | | |
| 0902.0431_FO1397 | 71 | 1.000 | \alpha \in \operatorname{Hom}_{C}\left(\mathfrak{J}^{C}\right) | | |
| 0902.0431_FO1398 | 71 | 1.000 | \alpha^{*}:\left\langle\alpha^{*} X, Y\right\rangle=\langle X, \alpha Y\rangle | | |
| 0902.0431_FO1399 | 71 | 1.000 | \alpha \in F_{4}{ }^{C} | | |
| 0902.0431_FO1400 | 71 | 1.000 | \alpha^{*}=\tau \alpha^{-1} \tau \in F_{4}{ }^{C} | | |
| 0902.0431_FO1401 | 71 | 0.999 | \alpha^{C}: \mathfrak{J}^{C} \rightarrow \mathfrak{J}^{C} | | |
| 0902.0431_FO1402 | 71 | 0.998 | \alpha^{C} \in F_{4}{ }^{C} | | |
| 0902.0431_FO1403 | 71 | 0.998 | F_{4}{ }^{C}: F_{4} \subset F_{4}{ }^{C} | | |
| 0902.0431_FO1404 | 71 | 1.000 | \left\langle\alpha^{*} X, Y\right\rangle=\langle X, \alpha Y\rangle=(\tau X, \alpha Y)=\left(\alpha^{-1} \tau X, Y\right)=\left\langle\tau \alpha^{-1} \tau X, Y\right\rangle | | |
| 0902.0431_FO1405 | 71 | 1.000 | X, Y \in \mathfrak{J}^{C} | | |
| 0902.0431_FO1406 | 71 | 0.999 | \tau \alpha X=\alpha \tau X=\alpha X | | |
| 0902.0431_FO1407 | 71 | 1.000 | \alpha X \in \mathfrak{J} | | |
| 0902.0431_FO1408 | 71 | 1.000 | \alpha^{\prime} \in F_{4} | | |
| 0902.0431_FO1409 | 72 | 0.999 | \operatorname{Iso}_{C}\left(\mathfrak{J}^{C}\right)=G L(27, C) | | |
| 0902.0431_FO1410 | 72 | 1.000 | \alpha^{*} \in F_{4}{ }^{C} | | |
| 0902.0431_FO1411 | 72 | 1.000 | U\left(\mathfrak{J}^{C}\right)=\left\{\alpha \in \operatorname{Iso}_{C}\left(\mathfrak{J}^{C}\right) \mid\langle\alpha X, \alpha Y\rangle=\langle X, Y\rangle\right\} | | |
| 0902.0431_FO1412 | 72 | 1.000 | d=\operatorname{dim} F_{4}{ }^{C}-\operatorname{dim} F_{4}= | | |
| 0902.0431_FO1413 | 72 | 1.000 | 2 \times 52-52=52 | | |
| 0902.0431_FO1414 | 72 | 1.000 | I_{1}=\operatorname{diag}(-1,1,1) \in M(3, \boldsymbol{R}) | | |
| 0902.0431_FO1415 | 72 | 1.000 | \mathfrak{J}\left(3, \mathfrak{C}^{\prime}\right) | | |
| 0902.0431_FO1416 | 72 | 1.000 | \mathfrak{J}(1,2, \mathfrak{C}) | | |
| 0902.0431_FO1417 | 72 | 0.998 | \mathfrak{J}(1 | | |
| 0902.0431_FO1418 | 72 | 1.000 | 2, \mathfrak{C}) | | |
| 0902.0431_FO1419 | 72 | 1.000 | z\left(F_{4(4)}\right) | | |
| 0902.0431_FO1420 | 72 | 1.000 | z\left(F_{4(-20)}\right) | | |
| 0902.0431_FO1421 | 73 | 1.000 | \lambda | | |
| 0902.0431_FO1422 | 73 | 1.000 | \phi \in \mathfrak{e}_{6}{ }^{C} | | |
| 0902.0431_FO1423 | 74 | 0.539 | \phi \in \mathfrak{e}_{6}^{C} | | |
| 0902.0431_FO1424 | 74 | 0.539 | T=\phi E | | |
| 0902.0431_FO1425 | 74 | 0.539 | T \in \mathfrak{J}^{C} | | |
| 0902.0431_FO1426 | 74 | 0.539 | \operatorname{tr}(T)=0 | | |
| 0902.0431_FO1427 | 74 | 1.000 | \operatorname{tr}(T)=(T, E, E)=(\phi E, E, E)=0 | | |
| 0902.0431_FO1428 | 74 | 1.000 | \delta=\phi-\widetilde{T} | | |
| 0902.0431_FO1429 | 74 | 1.000 | \delta \in \mathfrak{e}_{6}{ }^{C} | | |
| 0902.0431_FO1430 | 74 | 0.999 | \delta E=\phi E-\widetilde{T} E=T-T=0 | | |
| 0902.0431_FO1431 | 74 | 0.999 | \phi=\delta+\widetilde{T} | | |
| 0902.0431_FO1432 | 74 | 1.000 | T=0 | | |
| 0902.0431_FO1433 | 74 | 1.000 | \delta=0 | | |
| 0902.0431_FO1434 | 74 | 1.000 | \operatorname{dim}_{C}\left(\mathfrak{e}_{6}{ }^{C}\right)=52+26=78 | | |
| 0902.0431_FO1435 | 74 | 0.642 | \left[\phi_{1}, \phi_{2}\right] | | |
| 0902.0431_FO1436 | 74 | 1.000 | \phi_{i}=\delta_{i}+\widetilde{T}_{i}, \delta_{i} \in \mathfrak{f}_{4}{ }^{C}, T_{i} \in \mathfrak{J}_{0}{ }^{C} | | |
| 0902.0431_FO1437 | 74 | 1.000 | [\delta, \widetilde{T}]=\widetilde{\delta T} | | |
| 0902.0431_FO1438 | 74 | 1.000 | \delta \in \mathfrak{f}_{4}{ }^{C}, T \in \mathfrak{J}_{0}{ }^{C} | | |
| 0902.0431_FO1439 | 74 | 0.997 | \phi=\delta+\widetilde{T} \in \mathfrak{e}_{6}{ }^{C}, \delta \in \mathfrak{f}_{4}{ }^{C}, T \in \mathfrak{J}_{0}{ }^{C} | | |
| 0902.0431_FO1440 | 74 | 1.000 | -{ }^{t} \phi \in \mathfrak{e}_{6}{ }^{C} | | |
| 0902.0431_FO1441 | 74 | 0.992 | \left(-{ }^{t} \phi X, Y\right)=-(X, \phi Y)=-(X, \delta Y+\widetilde{T} Y)=-(X, \delta Y)-(X, T \circ Y) | | |
| 0902.0431_FO1442 | 74 | 0.997 | -{ }^{t} \phi=\delta-\widetilde{T} \in \mathfrak{e}_{6}{ }^{C} | | |
| 0902.0431_FO1443 | 74 | 1.000 | \phi \in \mathfrak{e}_{6} | | |
| 0902.0431_FO1444 | 74 | 0.955 | \tau \lambda(\phi) \tau=\phi | | |
| 0902.0431_FO1445 | 74 | 0.955 | \phi | | |
| 0902.0431_FO1446 | 74 | 0.999 | \phi=\delta+\widetilde{T^{\prime}}, \delta \in \mathfrak{f}_{4}{ }^{C}, T^{\prime} \in \mathfrak{J}_{0}{ }^{C} | | |
| 0902.0431_FO1447 | 75 | 1.000 | \tau \delta \tau-\widetilde{\tau T^{\prime}}=\delta+\widetilde{T^{\prime}} | | |
| 0902.0431_FO1448 | 75 | 1.000 | \tau \delta \tau=\delta | | |
| 0902.0431_FO1449 | 75 | 1.000 | \tau T^{\prime}=-T^{\prime} | | |
| 0902.0431_FO1450 | 75 | 1.000 | T^{\prime} | | |
| 0902.0431_FO1451 | 75 | 1.000 | T^{\prime}=i T, T \in \mathfrak{J}_{0} | | |
| 0902.0431_FO1452 | 75 | 1.000 | \phi^{*} | | |
| 0902.0431_FO1453 | 75 | 0.997 | \phi^{*}=\tau^{t} \phi \tau \in \mathfrak{e}_{6}{ }^{C} | | |
| 0902.0431_FO1454 | 75 | 0.997 | \phi \in \mathfrak{e}_{6}{ }^{C}, \phi | | |
| 0902.0431_FO1455 | 75 | 0.999 | \phi^{*}=-\phi | | |
| 0902.0431_FO1456 | 75 | 0.997 | p: \mathfrak{e}_{6}{ }^{C} \rightarrow \mathfrak{f}_{4}{ }^{C} | | |
| 0902.0431_FO1457 | 75 | 0.997 | q: \mathfrak{e}_{6}{ }^{C} \rightarrow \widetilde{\mathfrak{J}}_{0}{ }^{C} | | |
| 0902.0431_FO1458 | 75 | 0.997 | \mathfrak{e}_{6}{ }^{C}=\mathfrak{f}_{4}{ }^{C} \oplus \widetilde{\mathfrak{J}}_{0}{ }^{C} | | |
| 0902.0431_FO1459 | 75 | 1.000 | \delta \in p(\mathfrak{a}) | | |
| 0902.0431_FO1460 | 75 | 1.000 | T \in \mathfrak{J}_{0}{ }^{C} | | |
| 0902.0431_FO1461 | 75 | 1.000 | \delta+\widetilde{T} \in \mathfrak{a} | | |
| 0902.0431_FO1462 | 75 | 1.000 | \delta_{1} \in \mathfrak{f}_{4}{ }^{C} | | |
| 0902.0431_FO1463 | 75 | 1.000 | \left[\delta_{1}, \delta\right] \in \mathfrak{a} | | |
| 0902.0431_FO1464 | 75 | 1.000 | \mathfrak{f}_{4}{ }^{C} \cap \mathfrak{a} \neq\{0\} | | |
| 0902.0431_FO1465 | 75 | 1.000 | \widetilde{\mathfrak{J}}_{0}{ }^{C} \cap \mathfrak{a} \neq\{0\} | | |
| 0902.0431_FO1466 | 75 | 1.000 | \mathfrak{f}_{4}{ }^{C} \cap \mathfrak{a}=\{0\} | | |
| 0902.0431_FO1467 | 75 | 0.876 | \widetilde{\mathfrak{J}}_{0}^{C} \cap \mathfrak{a}=\{0\} | | |
| 0902.0431_FO1468 | 75 | 0.876 | p \mid \mathfrak{a}: \mathfrak{a} \rightarrow \mathfrak{f}_{4}{ }^{C} | | |
| 0902.0431_FO1469 | 75 | 1.000 | p(\mathfrak{a})=\mathfrak{f}_{4}{ }^{C} | | |
| 0902.0431_FO1470 | 75 | 1.000 | \operatorname{dim}_{C} \mathfrak{a}=\operatorname{dim}_{C} p(\mathfrak{a})=\operatorname{dim}_{C} \mathfrak{f}_{4}{ }^{C}=52 | | |
| 0902.0431_FO1471 | 75 | 0.673 | q \mid \mathfrak{a}: \mathfrak{a} \rightarrow \widetilde{\mathfrak{J}}_{0}^{C} | | |
| 0902.0431_FO1472 | 75 | 0.673 | \operatorname{dim}_{C} \mathfrak{a} \leq \operatorname{dim}_{C} \widetilde{\mathfrak{J}}_{0}^{C}=\operatorname{dim}_{C} \mathfrak{J}_{0}{ }^{C}=26 | | |
| 0902.0431_FO1473 | 75 | 1.000 | \mathfrak{f}_{4}{ }^{C} \cap \mathfrak{a}=\mathfrak{f}_{4}{ }^{C} | | |
| 0902.0431_FO1474 | 75 | 1.000 | \mathfrak{a} \supset \mathfrak{f}_{4}{ }^{C} | | |
| 0902.0431_FO1475 | 75 | 0.896 | \mathfrak{a} \supset \mathfrak{f}_{4}{ }^{C} \oplus \tilde{\mathfrak{J}}_{0}{ }^{C}=\mathfrak{e}_{6}{ }^{C} | | |
| 0902.0431_FO1476 | 75 | 0.953 | \widetilde{\mathfrak{J}}_{0}^{C} \cap \mathfrak{a} \neq\{0\} | | |
| 0902.0431_FO1477 | 75 | 0.953 | \widetilde{A}\left(A \in \mathfrak{J}_{0}{ }^{C}\right) | | |
| 0902.0431_FO1478 | 75 | 0.953 | \widetilde{\mathfrak{J}}_{0}^{C} \cap \mathfrak{a} \subset \mathfrak{a} | | |
| 0902.0431_FO1479 | 75 | 0.999 | 0 \neq[\widetilde{A}, \widetilde{B}] \in \mathfrak{f}_{4}{ }^{C} \cap \mathfrak{a} | | |
| 0902.0431_FO1480 | 76 | 1.000 | \mathfrak{a}=\mathfrak{e}_{6}{ }^{C} | | |
| 0902.0431_FO1481 | 76 | 0.965 | \mathfrak{e}_{6}{ }^{C} \mathfrak{J}^{C}=\left\{\sum_{i} \phi_{i} A_{i} \mid \phi_{i} \in \mathfrak{e}_{6}{ }^{C}, A_{i} \in \mathfrak{J}^{C}\right\}=\mathfrak{J}^{C} | | |
| 0902.0431_FO1482 | 76 | 1.000 | X=X(\xi, x) | | |
| 0902.0431_FO1483 | 76 | 1.000 | \xi_{1} \neq 0 | | |
| 0902.0431_FO1484 | 76 | 1.000 | \xi_{1} E_{1}=\left(2 X \circ E_{1}-X\right) \circ E_{1} \in \mathfrak{a} | | |
| 0902.0431_FO1485 | 76 | 1.000 | E_{1} \in \mathfrak{a} | | |
| 0902.0431_FO1486 | 76 | 1.000 | F_{2}(1)=2 E_{1} \circ F_{2}(1) \in \mathfrak{a} | | |
| 0902.0431_FO1487 | 76 | 1.000 | E_{1}+E_{3}=F_{2}(1) \circ F_{2}(1) \in \mathfrak{a} | | |
| 0902.0431_FO1488 | 76 | 1.000 | E_{3}=\left(E_{1}+E_{3}\right)-E_{1} \in \mathfrak{a} | | |
| 0902.0431_FO1489 | 76 | 1.000 | E_{2} \in \mathfrak{a} | | |
| 0902.0431_FO1490 | 76 | 1.000 | E=E_{1}+E_{2}+E_{3} \in \mathfrak{a} | | |
| 0902.0431_FO1491 | 76 | 1.000 | X \in \mathfrak{J}^{C}, X=E \circ X \in \mathfrak{a} | | |
| 0902.0431_FO1492 | 76 | 1.000 | \mathfrak{a}=\mathfrak{J}^{C} | | |
| 0902.0431_FO1493 | 76 | 1.000 | \xi_{2} \neq 0 | | |
| 0902.0431_FO1494 | 76 | 1.000 | \xi_{3} \neq 0 | | |
| 0902.0431_FO1495 | 76 | 1.000 | \xi_{1}=\xi_{2}=\xi_{3}=0, x_{1} \neq 0 | | |
| 0902.0431_FO1496 | 76 | 1.000 | F_{1}\left(x_{1}\right)=4\left(X \circ E_{2}\right) \circ E_{3} \in \mathfrak{a} | | |
| 0902.0431_FO1497 | 76 | 1.000 | a \in \mathfrak{C}^{C} | | |
| 0902.0431_FO1498 | 76 | 1.000 | \left(x_{1}, a\right)=1 | | |
| 0902.0431_FO1499 | 76 | 1.000 | F_{1}\left(x_{1}\right) \circ F_{1}(a)=\left(x_{1}, a\right)\left(E_{2}+E_{3}\right)=E_{2}+E_{3} \in \mathfrak{a} | | |
| 0902.0431_FO1500 | 76 | 1.000 | X \in W | | |
| 0902.0431_FO1501 | 76 | 0.598 | \mathfrak{J}^{C}\left((1)\right. | | |
| 0902.0431_FO1502 | 76 | 0.598 | W=\mathfrak{J}^{C} | | |
| 0902.0431_FO1503 | 76 | 1.000 | \mathfrak{e}_{6}{ }^{C} \mathfrak{J}^{C} | | |
| 0902.0431_FO1504 | 76 | 0.925 | \mathfrak{e}_{6}{ }^{C} \mathfrak{J}^{C}=\mathfrak{J}^{C} | | |
| 0902.0431_FO1505 | 76 | 0.998 | A \vee B \in \mathfrak{e}_{6}{ }^{C} | | |
| 0902.0431_FO1506 | 76 | 1.000 | X \circ(X \times X)=(\operatorname{det} X) E, X \in \mathfrak{J}^{C} | | |
| 0902.0431_FO1507 | 76 | 1.000 | \lambda A+\mu B+\nu X | | |
| 0902.0431_FO1508 | 77 | 1.000 | E_{i} \vee E_{j}=0, i \neq j | | |
| 0902.0431_FO1509 | 77 | 1.000 | E_{1} \vee E_{1}=\frac{1}{3}\left(2 E_{1}-E_{2}-E_{3}\right)^{\sim} | | |
| 0902.0431_FO1510 | 77 | 0.988 | \phi \in \operatorname{Hom}_{C}\left(\mathfrak{J}^{C}\right) | | |
| 0902.0431_FO1511 | 77 | 0.988 | -{ }^{t} \phi | | |
| 0902.0431_FO1512 | 77 | 0.988 | \phi^{\prime} | | |
| 0902.0431_FO1513 | 78 | 1.000 | (A \vee B)^{\prime}=\left([\widetilde{A}, \widetilde{B}]+\left(A \circ B-\frac{1}{3}(A, B) E\right)^{\sim}\right)^{\prime}=[\widetilde{A}, \widetilde{B}]-\left(A \circ B-\frac{1}{3}(A, B) E\right)^{\sim} | | |
| 0902.0431_FO1514 | 78 | 0.999 | ( | | |
| 0902.0431_FO1515 | 78 | 0.999 | )=-[\widetilde{B}, \widetilde{A}]-\left(A \circ B-\frac{1}{3}(A, B) E\right)^{\sim}=-B \vee A | | |
| 0902.0431_FO1516 | 78 | 0.869 | \phi=\sum_{i}\left(A_{i} \vee B_{i}\right), A_{i}, B_{i} \in \mathfrak{J}^{C} | | |
| 0902.0431_FO1517 | 78 | 1.000 | [\phi, A \vee B] X=\phi(A \vee B) X-(A \vee B) \phi X | | |
| 0902.0431_FO1518 | 78 | 0.988 | \mathfrak{a}=\left\{\sum_{i}\left(A_{i} \vee B_{i}\right) \mid A_{i}, B_{i} \in \mathfrak{J}^{C}\right\} | | |
| 0902.0431_FO1519 | 78 | 1.000 | \left(\phi_{1}, \phi_{2}\right)_{6} | | |
| 0902.0431_FO1520 | 78 | 0.749 | \phi \in \mathfrak{e}_{6}^{C}, A, B \in \mathfrak{J}^{C} | | |
| 0902.0431_FO1521 | 78 | 0.993 | \phi=\delta+\widetilde{T}, \phi_{i}=\delta_{i}+\widetilde{T}_{i}, \delta, \delta_{i} \in \mathfrak{f}_{4}{ }^{C}, T, T_{i} \in \mathfrak{J}_{0}{ }^{C} | | |
| 0902.0431_FO1522 | 79 | 1.000 | \phi=\delta+\widetilde{T}, \delta \in \mathfrak{f}_{4}{ }^{C}, T \in \mathfrak{J}_{0}{ }^{C} | | |
| 0902.0431_FO1523 | 79 | 1.000 | B_{6} | | |
| 0902.0431_FO1524 | 79 | 1.000 | \phi=\phi_{1}=\phi_{2}=\left(E_{1}-E_{2}\right)^{\sim} | | |
| 0902.0431_FO1525 | 79 | 1.000 | (\operatorname{ad} \phi)^{2} | | |
| 0902.0431_FO1526 | 80 | 0.877 | =0 | | |
| 0902.0431_FO1527 | 80 | 0.998 | k=12 | | |
| 0902.0431_FO1528 | 80 | 0.998 | \operatorname{tr}(\phi \phi) | | |
| 0902.0431_FO1529 | 80 | 1.000 | k^{\prime}=4 | | |
| 0902.0431_FO1530 | 80 | 1.000 | A \vee(A \times A)=0, \quad A \in \mathfrak{J}^{C} | | |
| 0902.0431_FO1531 | 80 | 0.999 | A \in \mathfrak{J}^{C}, A \neq 0 | | |
| 0902.0431_FO1532 | 80 | 0.999 | B \in \mathfrak{J}^{C} | | |
| 0902.0431_FO1533 | 80 | 0.999 | A \vee B \neq 0 | | |
| 0902.0431_FO1534 | 80 | 0.990 | (\phi,(A \times A) \vee A)_{6}=(\phi(A \times A), A) | | |
| 0902.0431_FO1535 | 80 | 0.867 | (\phi,(A \times A) \vee A)_{6}=0 | | |
| 0902.0431_FO1536 | 80 | 0.867 | (A \times A) \vee A=0 | | |
| 0902.0431_FO1537 | 80 | 1.000 | A \vee(A \times A)=0 | | |
| 0902.0431_FO1538 | 80 | 1.000 | \lambda A+\mu B+\nu C | | |
| 0902.0431_FO1539 | 80 | 1.000 | A \vee B=0 | | |
| 0902.0431_FO1540 | 80 | 1.000 | B \vee A=0 | | |
| 0902.0431_FO1541 | 80 | 1.000 | \phi \in \mathfrak{e}_{6}{ }^{C}, 0=(\phi, B \vee A)_{6}=(\phi B, A) | | |
| 0902.0431_FO1542 | 80 | 1.000 | \mathfrak{e}_{6}{ }^{C} \mathfrak{J}^{C}= | | |
| 0902.0431_FO1543 | 80 | 1.000 | \left(\mathfrak{J}^{C}, A\right)=0 | | |
| 0902.0431_FO1544 | 81 | 1.000 | \left(\mathfrak{M}^{r}\right)^{C} | | |
| 0902.0431_FO1545 | 81 | 1.000 | \delta=\left(\delta_{1}, \delta_{2}, \delta_{3}\right) \in \mathfrak{d}_{4}{ }^{C} | | |
| 0902.0431_FO1546 | 81 | 1.000 | \left(\delta_{1} x\right) y+ | | |
| 0902.0431_FO1547 | 81 | 0.998 | x\left(\delta_{2} y\right)=\overline{\delta_{3}(\overline{x y})}, x, y \in \mathfrak{C}^{C} | | |
| 0902.0431_FO1548 | 81 | 0.998 | \delta:\left(\mathfrak{M}^{r}\right)^{C} \rightarrow\left(\mathfrak{M}^{r}\right)^{C} | | |
| 0902.0431_FO1549 | 81 | 1.000 | \delta: \mathfrak{J}^{C} \rightarrow \mathfrak{J}^{C} | | |
| 0902.0431_FO1550 | 81 | 1.000 | \delta \in \mathfrak{d}_{4}{ }^{C} | | |
| 0902.0431_FO1551 | 81 | 0.958 | (i, i) | | |
| 0902.0431_FO1552 | 81 | 0.958 | \delta X \circ Y+Y \circ \delta Y | | |
| 0902.0431_FO1553 | 81 | 1.000 | \delta X \circ Y+Y \circ \delta Y(i \neq j) | | |
| 0902.0431_FO1554 | 81 | 1.000 | i, j, k | | |
| 0902.0431_FO1555 | 81 | 0.999 | i=k \neq j | | |
| 0902.0431_FO1556 | 81 | 0.999 | i \neq k=j | | |
| 0902.0431_FO1557 | 81 | 0.999 | \sigma_{l l}=0 | | |
| 0902.0431_FO1558 | 81 | 1.000 | x_{l l} \in C | | |
| 0902.0431_FO1559 | 81 | 1.000 | T \in M\left(3, \mathfrak{C}^{C}\right) | | |
| 0902.0431_FO1560 | 81 | 1.000 | \widetilde{T}: \mathfrak{J}^{C} \rightarrow \mathfrak{J}^{C} | | |
| 0902.0431_FO1561 | 82 | 0.995 | T \in M\left(3, \mathfrak{C}^{C}\right), \operatorname{tr}(T)=0 | | |
| 0902.0431_FO1562 | 82 | 0.995 | \widetilde{T} \in \mathfrak{e}_{6}{ }^{C} | | |
| 0902.0431_FO1563 | 82 | 0.920 | T=T_{1}+T_{2}, T_{1}=\frac{T+T^{*}}{2}, T_{2}=\frac{T-T^{*}}{2} | | |
| 0902.0431_FO1564 | 82 | 0.920 | \widetilde{T}_{1} \in \mathfrak{e}_{6}{ }^{C} | | |
| 0902.0431_FO1565 | 82 | 1.000 | \widetilde{T}_{2} \in \mathfrak{f}_{4}{ }^{C} | | |
| 0902.0431_FO1566 | 82 | 1.000 | \subset \mathfrak{e}_{6}{ }^{C} | | |
| 0902.0431_FO1567 | 82 | 1.000 | \widetilde{T}=\widetilde{T}_{1}+\widetilde{T}_{2} \in | | |
| 0902.0431_FO1568 | 82 | 0.777 | \delta \in \mathfrak{d}_{4}^{C} | | |
| 0902.0431_FO1569 | 82 | 0.777 | R \in\left(\mathfrak{M}^{r}\right)^{C}, \operatorname{tr}(R)=0 | | |
| 0902.0431_FO1570 | 82 | 1.000 | H \in M(3, C), \operatorname{tr}(H)=0 | | |
| 0902.0431_FO1571 | 82 | 0.999 | (\widetilde{\delta R}) X=\delta R \circ X=\delta(R \circ X)-R \circ \delta X | | |
| 0902.0431_FO1572 | 82 | 0.999 | =\delta(\widetilde{R} X)- | | |
| 0902.0431_FO1573 | 82 | 0.999 | \widetilde{R}(\delta X)=[\delta, \widetilde{R}] X, X \in \mathfrak{J}^{C} | | |
| 0902.0431_FO1574 | 82 | 0.999 | \widetilde{\delta R}=[\delta, \widetilde{R}] | | |
| 0902.0431_FO1575 | 82 | 1.000 | [\widetilde{H}, \widetilde{T}] X=\widetilde{H} \widetilde{T} X-\widetilde{T} \widetilde{H} X | | |
| 0902.0431_FO1576 | 82 | 1.000 | H \in M(3, C) | | |
| 0902.0431_FO1577 | 83 | 1.000 | \mu_{1}+\mu_{2}+\mu_{3}=0 | | |
| 0902.0431_FO1578 | 83 | 1.000 | \pm \lambda_{k} \pm \lambda_{l} | | |
| 0902.0431_FO1579 | 83 | 1.000 | \mathfrak{d}_{4}{ }^{C}\left(\subset \mathfrak{f}_{4}{ }^{C} \subset \mathfrak{e}_{6}{ }^{C}\right) | | |
| 0902.0431_FO1580 | 83 | 1.000 | S \in \delta_{4}{ }^{C} \subset \mathfrak{e}_{6}{ }^{C} | | |
| 0902.0431_FO1581 | 83 | 1.000 | a E_{k l} \in M\left(3, \mathfrak{C}^{C}\right) | | |
| 0902.0431_FO1582 | 83 | 1.000 | F_{k l}(a): F_{k l}(a)=a E_{k l}, a \in \mathfrak{C}^{C}, k \neq l | | |
| 0902.0431_FO1583 | 83 | 1.000 | F_{23}(a)=\left(\begin{array}{lll}0 & 0 & 0 \\ 0 & 0 & a \\ 0 & 0 & 0\end{array}\right) | | |
| 0902.0431_FO1584 | 83 | 1.000 | a=e_{k}+i e_{4+k} | | |
| 0902.0431_FO1585 | 83 | 1.000 | h_{\delta} a=h_{\delta}\left(e_{k}+i e_{4+k}\right)=\lambda_{k}\left(e_{k}+i e_{4+k}\right)=\lambda_{k} a | | |
| 0902.0431_FO1586 | 83 | 0.997 | \left.\left[H, F_{23}(a)\right]=H F_{23}(a)-F_{23}(a) H=\left(\mu_{2}-\mu_{3}\right) F_{23}(a)\right) | | |
| 0902.0431_FO1587 | 83 | 0.997 | \lambda_{k}+\frac{1}{2}\left(\mu_{2}-\mu_{3}\right) | | |
| 0902.0431_FO1588 | 84 | 1.000 | \widetilde{F}_{23}\left(e_{k}+i e_{4+k}\right) | | |
| 0902.0431_FO1589 | 84 | 1.000 | -\lambda_{k}+\frac{1}{2}\left(\mu_{2}-\mu_{3}\right) | | |
| 0902.0431_FO1590 | 84 | 1.000 | \widetilde{F}_{23}\left(e_{k}-i e_{4+k}\right) | | |
| 0902.0431_FO1591 | 84 | 0.999 | -\lambda_{k}+\frac{1}{2}\left(\mu_{3}-\mu_{2}\right) | | |
| 0902.0431_FO1592 | 84 | 0.999 | \lambda_{k}+\frac{1}{2}\left(\mu_{3}-\mu_{2}\right) | | |
| 0902.0431_FO1593 | 84 | 0.999 | \widetilde{F}_{32}\left(e_{k}-i e_{4+k}\right) | | |
| 0902.0431_FO1594 | 85 | 1.000 | n_{1} n_{2} \cdots n_{6} | | |
| 0902.0431_FO1595 | 85 | 1.000 | n_{1} \alpha_{1}+n_{2} \alpha_{2}+ | | |
| 0902.0431_FO1596 | 85 | 0.578 | \cdots+n_{6} \alpha_{6} | | |
| 0902.0431_FO1597 | 86 | 1.000 | \Pi=\left\{\alpha_{1}, \alpha_{2}, \cdots, \alpha_{6}\right\} | | |
| 0902.0431_FO1598 | 86 | 0.909 | h=\sum_{k=0}^{3} \lambda_{k} H_{k}+\left(\sum_{j=1}^{3} \mu_{j} E_{j}\right)^{\sim}, h^{\prime}=\sum_{k=0}^{3} \lambda_{k}{ }^{\prime} H_{k}+\left(\sum_{j=1}^{3} \mu_{j}{ }^{\prime} E_{j}\right)^{\sim} \in \mathfrak{h}_{\boldsymbol{R}} | | |
| 0902.0431_FO1599 | 86 | 1.000 | \alpha_{i}\left(B_{6}\left(H_{\alpha}, H\right)=\alpha(H), H \in\right. | | |
| 0902.0431_FO1600 | 87 | 1.000 | \alpha_{1}, \alpha_{2}, \cdots, \alpha_{6} | | |
| 0902.0431_FO1601 | 87 | 1.000 | T \oplus D_{5} | | |
| 0902.0431_FO1602 | 87 | 1.000 | C_{1} \oplus A_{5} | | |
| 0902.0431_FO1603 | 87 | 1.000 | A_{2} \oplus A_{2} \oplus A_{2} | | |
| 0902.0431_FO1604 | 87 | 1.000 | \tau, \tau \gamma | | |
| 0902.0431_FO1605 | 87 | 1.000 | C_{4} | | |
| 0902.0431_FO1606 | 87 | 1.000 | \left(E_{6}\right)^{\tau} | | |
| 0902.0431_FO1607 | 87 | 1.000 | \alpha \in E_{6} | | |
| 0902.0431_FO1608 | 87 | 1.000 | (\alpha X, \alpha Y)=\langle\tau \alpha X, \alpha Y\rangle=\langle\alpha \tau X, \alpha Y\rangle=\langle\tau X, Y\rangle= | | |
| 0902.0431_FO1609 | 87 | 1.000 | (\alpha X, \alpha Y)=(X, Y) | | |
| 0902.0431_FO1610 | 87 | 0.999 | \quad\left(E_{6}\right)^{\tau}=\left(E_{6}\right)_{E} \cong F_{4} | | |
| 0902.0431_FO1611 | 88 | 1.000 | \left(E_{6}\right)^{\tau} \cong F_{4} | | |
| 0902.0431_FO1612 | 88 | 1.000 | \alpha \in\left(E_{6}\right)^{\tau} | | |
| 0902.0431_FO1613 | 88 | 1.000 | \alpha \mid \mathfrak{J} | | |
| 0902.0431_FO1614 | 88 | 1.000 | \alpha^{C}: \mathfrak{J}^{C} \rightarrow \mathfrak{J}^{C}, \alpha^{C}\left(X_{1}+i X_{2}\right)=\alpha X_{1}+ | | |
| 0902.0431_FO1615 | 88 | 1.000 | i \alpha X_{2} | | |
| 0902.0431_FO1616 | 88 | 1.000 | F_{4} \ni \alpha \rightarrow \alpha^{C} \in\left(E_{6}\right)^{\tau} | | |
| 0902.0431_FO1617 | 88 | 0.959 | \left(E_{6}\right)_{0} | | |
| 0902.0431_FO1618 | 88 | 1.000 | \alpha_{12}(t): \mathfrak{J}^{C} \rightarrow \mathfrak{J}^{C} | | |
| 0902.0431_FO1619 | 88 | 1.000 | \alpha_{12}(t) \in\left(E_{6}\right)_{0} | | |
| 0902.0431_FO1620 | 88 | 1.000 | \alpha_{13}(t), \alpha_{23}(t) \in\left(E_{6}\right)_{0} | | |
| 0902.0431_FO1621 | 88 | 1.000 | \alpha_{1}(a): \mathfrak{J}^{C} \rightarrow \mathfrak{J}^{C} | | |
| 0902.0431_FO1622 | 88 | 1.000 | \alpha_{1}(a) X(\xi, x)=Y(\eta, y) | | |
| 0902.0431_FO1623 | 88 | 1.000 | \alpha_{1}(a) \in\left(E_{6}\right)_{0} | | |
| 0902.0431_FO1624 | 88 | 0.998 | E_{1}-E_{2} \in \mathfrak{J}_{0} | | |
| 0902.0431_FO1625 | 88 | 0.998 | i\left(E_{1}-E_{2}\right)^{\sim} \in \mathfrak{e}_{6} | | |
| 0902.0431_FO1626 | 88 | 1.000 | \alpha_{12}(t)=\exp i t\left(E_{1}-E_{2}\right)^{\sim} | | |
| 0902.0431_FO1627 | 88 | 1.000 | F_{1}(a) \in \mathfrak{J}_{0} | | |
| 0902.0431_FO1628 | 88 | 1.000 | i \widetilde{F}_{1}(a) \in \mathfrak{e}_{6} | | |
| 0902.0431_FO1629 | 88 | 1.000 | \alpha_{1}(a)= | | |
| 0902.0431_FO1630 | 88 | 1.000 | \exp i \widetilde{F}_{1}(a) | | |
| 0902.0431_FO1631 | 88 | 1.000 | X \in \mathfrak{J}^{C} | | |
| 0902.0431_FO1632 | 88 | 1.000 | \alpha \in\left(E_{6}\right)_{0} | | |
| 0902.0431_FO1633 | 89 | 1.000 | |\xi| | | |
| 0902.0431_FO1634 | 89 | 1.000 | \sqrt{\xi(\tau \xi)} | | |
| 0902.0431_FO1635 | 89 | 1.000 | \xi \in \boldsymbol{R}^{C}=C | | |
| 0902.0431_FO1636 | 89 | 1.000 | \mathfrak{X}=\left\{\alpha X \mid \alpha \in\left(E_{6}\right)_{0}\right\} | | |
| 0902.0431_FO1637 | 89 | 1.000 | \left|\xi_{1}\right|^{2}+\left|\xi_{2}\right|^{2}+\left|\xi_{3}\right|^{2} | | |
| 0902.0431_FO1638 | 89 | 1.000 | \left|\eta_{1}\right|^{2}+\left|\eta_{2}\right|^{2}+\left|\eta_{3}\right|^{2} | | |
| 0902.0431_FO1639 | 89 | 1.000 | \xi_{2}, \xi_{3} | | |
| 0902.0431_FO1640 | 89 | 1.000 | \alpha_{12}\left(t_{1}\right) | | |
| 0902.0431_FO1641 | 89 | 1.000 | \alpha_{13}\left(t_{2}\right) | | |
| 0902.0431_FO1642 | 89 | 1.000 | q \neq 0 | | |
| 0902.0431_FO1643 | 89 | 1.000 | a(t)=\frac{q}{|q|} t, t>0 | | |
| 0902.0431_FO1644 | 89 | 1.000 | \alpha_{1}(a(t)) \in\left(E_{6}\right)_{0} | | |
| 0902.0431_FO1645 | 89 | 1.000 | i \frac{\left(a(t), x_{1}\right)}{|a(t)|}=i \nu-|q| | | |
| 0902.0431_FO1646 | 89 | 1.000 | \nu=\left(\frac{q}{|q|}, p\right) | | |
| 0902.0431_FO1647 | 89 | 1.000 | Y(\eta(t), y(t))=\alpha_{1}(a(t)) X_{0} \in \mathfrak{X} | | |
| 0902.0431_FO1648 | 89 | 1.000 | q=0 | | |
| 0902.0431_FO1649 | 89 | 1.000 | p \neq 0 | | |
| 0902.0431_FO1650 | 89 | 1.000 | a(t)=\frac{p}{|p|} t, t>0 | | |
| 0902.0431_FO1651 | 89 | 1.000 | \beta_{1}(a(t)) \in F_{4} \subset\left(E_{6}\right)_{0} | | |
| 0902.0431_FO1652 | 89 | 1.000 | \left|\eta_{1}(t)\right|^{2}+ | | |
| 0902.0431_FO1653 | 89 | 1.000 | \left|\eta_{2}(t)\right|^{2}+\left|\eta_{3}(t)\right|^{2} | | |
| 0902.0431_FO1654 | 89 | 1.000 | \left|\xi_{1}\right|^{2}+\left|\xi_{2}\right|^{2}+\left|\xi_{3}\right|^{2}+2|p|^{2} | | |
| 0902.0431_FO1655 | 89 | 0.999 | \alpha_{2}(a), \alpha_{3}(a) \in\left(E_{6}\right)_{0} | | |
| 0902.0431_FO1656 | 89 | 0.999 | \alpha_{1}(a) | | |
| 0902.0431_FO1657 | 90 | 0.999 | E I V | | |
| 0902.0431_FO1658 | 90 | 0.999 | \quad E_{6} / F_{4} \simeq E I V | | |
| 0902.0431_FO1659 | 90 | 1.000 | X \in E I V | | |
| 0902.0431_FO1660 | 90 | 1.000 | \alpha X \in E I V | | |
| 0902.0431_FO1661 | 90 | 1.000 | E \in E I V | | |
| 0902.0431_FO1662 | 90 | 1.000 | X \in E I V \subset \mathfrak{J}^{C} | | |
| 0902.0431_FO1663 | 90 | 0.709 | \xi_{1}=\xi_{2}=\xi_{3}=1 | | |
| 0902.0431_FO1664 | 90 | 0.709 | 0 \leq\left(\xi_{2}-\xi_{3}\right)^{2}={\xi_{2}}^{2}+{\xi_{3}}^{2}-2{\xi_{2}} \xi_{3}= | | |
| 0902.0431_FO1665 | 90 | 0.993 | 3-\xi_{1}{ }^{2}-\frac{2}{\xi_{1}}=-\frac{\xi_{1}{ }^{3}-3 \xi_{1}+2}{\xi_{1}}=-\frac{\left(\xi_{1}-1\right)^{2}\left(\xi_{1}+2\right)}{\xi_{1}} \leq 0 | | |
| 0902.0431_FO1666 | 90 | 0.993 | \xi_{1}=1 | | |
| 0902.0431_FO1667 | 90 | 1.000 | \xi_{2}=\xi_{3}=1 | | |
| 0902.0431_FO1668 | 90 | 1.000 | \alpha X=E | | |
| 0902.0431_FO1669 | 90 | 0.999 | E I V=\left(E_{6}\right)_{0} E, E I V | | |
| 0902.0431_FO1670 | 90 | 1.000 | E_{6} / F_{4} \simeq E I V | | |
| 0902.0431_FO1671 | 90 | 1.000 | \alpha \in z\left(E_{6}\right) | | |
| 0902.0431_FO1672 | 90 | 1.000 | \beta \in F_{4} \subset E_{6} | | |
| 0902.0431_FO1673 | 90 | 1.000 | \beta \alpha E=\alpha \beta E=\alpha E | | |
| 0902.0431_FO1674 | 90 | 1.000 | \alpha E=Y=Y(\eta, y) \in \mathfrak{J}^{C} | | |
| 0902.0431_FO1675 | 91 | 1.000 | T=\left(\begin{array}{ccc}1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{array}\right),\left(\begin{array}{ccc}-1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1\end{array}\right) | | |
| 0902.0431_FO1676 | 91 | 1.000 | \left(\begin{array}{ccc}0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0\end{array}\right) \in S O(3) | | |
| 0902.0431_FO1677 | 91 | 1.000 | y_{1}=y_{2}=y_{3}=0 | | |
| 0902.0431_FO1678 | 91 | 1.000 | \eta_{1}=\eta_{2}=\eta_{3}(=\omega) | | |
| 0902.0431_FO1679 | 91 | 1.000 | \omega^{3}=\operatorname{det}(\alpha E)=\operatorname{det} E=1 | | |
| 0902.0431_FO1680 | 91 | 1.000 | \omega 1 \in z\left(E_{6}\right) | | |
| 0902.0431_FO1681 | 91 | 1.000 | \omega^{-1} \alpha \in z\left(E_{6}\right) | | |
| 0902.0431_FO1682 | 91 | 1.000 | \omega^{-1} \alpha E=E | | |
| 0902.0431_FO1683 | 91 | 1.000 | \omega^{-1} \alpha \in z\left(F_{4}\right) | | |
| 0902.0431_FO1684 | 91 | 1.000 | z\left(F_{4}\right)=\{1\} | | |
| 0902.0431_FO1685 | 91 | 1.000 | \omega^{-1} \alpha=1 | | |
| 0902.0431_FO1686 | 91 | 1.000 | \alpha=\omega 1 | | |
| 0902.0431_FO1687 | 91 | 1.000 | E_{6}=\left\{\alpha \in \operatorname{Iso}_{C}\left(\mathfrak{J}^{C}\right) \mid \operatorname{det}(\alpha X)=\operatorname{det} X,\langle\alpha X, \alpha Y\rangle=\langle X, Y\rangle\right\} | | |
| 0902.0431_FO1688 | 91 | 1.000 | \sigma: \mathfrak{J}^{C} \rightarrow \mathfrak{J}^{C} | | |
| 0902.0431_FO1689 | 91 | 1.000 | \sigma \in E_{6} | | |
| 0902.0431_FO1690 | 91 | 1.000 | \left(E_{6}\right)^{\sigma} | | |
| 0902.0431_FO1691 | 91 | 1.000 | \left(\mathfrak{J}^{C}\right)_{\sigma} | | |
| 0902.0431_FO1692 | 91 | 1.000 | \left(\mathfrak{J}^{C}\right)_{-\sigma} | | |
| 0902.0431_FO1693 | 91 | 0.672 | \mathfrak{E}_{1}{ }^{C}=\left\{\xi E_{1} \mid \xi \in C\right\} | | |
| 0902.0431_FO1694 | 91 | 0.672 | \mathfrak{J}^{C}=\left(\mathfrak{J}^{C}\right)_{\sigma} \oplus\left(\mathfrak{J}^{C}\right)_{-\sigma} | | |
| 0902.0431_FO1695 | 91 | 0.999 | \left(\mathfrak{J}^{C}\right)_{\sigma},\left(\mathfrak{J}^{C}\right)_{-\sigma} | | |
| 0902.0431_FO1696 | 91 | 1.000 | \alpha \in\left(E_{6}\right)^{\sigma} | | |
| 0902.0431_FO1697 | 91 | 1.000 | \xi \in C | | |
| 0902.0431_FO1698 | 92 | 1.000 | \alpha E_{1} \in \mathfrak{J}\left(2, \mathfrak{C}^{C}\right) | | |
| 0902.0431_FO1699 | 92 | 1.000 | \alpha E=\alpha E_{1}+\alpha E_{2}+\alpha E_{3} \in \mathfrak{J}\left(2, \mathfrak{C}^{C}\right) | | |
| 0902.0431_FO1700 | 92 | 1.000 | \alpha E=\xi_{2} E_{2}+\xi_{3} E_{3}+F_{1}\left(x_{1}\right), \xi_{2}, \xi_{3} \in C, x_{1} \in \mathfrak{C}^{C} | | |
| 0902.0431_FO1701 | 92 | 1.000 | \alpha E=0 | | |
| 0902.0431_FO1702 | 92 | 1.000 | \alpha E_{1} | | |
| 0902.0431_FO1703 | 92 | 0.999 | \alpha E_{1} \times \alpha E_{1}=\tau \alpha \tau\left(E_{1} \times E_{1}\right)=0 | | |
| 0902.0431_FO1704 | 92 | 1.000 | (\tau \xi) \xi=1 | | |
| 0902.0431_FO1705 | 92 | 1.000 | \left(E_{6}\right)_{E_{1}} | | |
| 0902.0431_FO1706 | 92 | 1.000 | \left(E_{6}\right)^{\sigma}:\left(E_{6}\right)_{E_{1}} \subset\left(E_{6}\right)^{\sigma} | | |
| 0902.0431_FO1707 | 92 | 0.926 | \left(\mathfrak{J}^{C}\right)_{\sigma}=\left\{X \in \mathfrak{J}^{C} \mid 4 E_{1} \times\left(E_{1} \times X\right)=X\right\} \oplus \mathfrak{E}_{1}^{C} | | |
| 0902.0431_FO1708 | 92 | 0.926 | \left(\mathfrak{J}^{C}\right)_{-\sigma}= | | |
| 0902.0431_FO1709 | 92 | 1.000 | \left\{X \in \mathfrak{J}^{C} \mid E_{1} \times X=0,\left\langle E_{1}, X\right\rangle=0\right\} | | |
| 0902.0431_FO1710 | 92 | 1.000 | \alpha \in\left(E_{6}\right)_{E_{1}} | | |
| 0902.0431_FO1711 | 92 | 1.000 | \sigma \alpha=\alpha \sigma | | |
| 0902.0431_FO1712 | 93 | 1.000 | V^{10} | | |
| 0902.0431_FO1713 | 93 | 0.835 | \quad\left(E_{6}\right)_{E_{1}} / \operatorname{Spin}(9) \simeq S^{9} | | |
| 0902.0431_FO1714 | 93 | 1.000 | S^{9}=\left\{X \in V^{10} \mid\langle X, X\rangle=2\right\} | | |
| 0902.0431_FO1715 | 93 | 1.000 | X \in S^{9} | | |
| 0902.0431_FO1716 | 93 | 1.000 | \alpha X \in S^{9} | | |
| 0902.0431_FO1717 | 93 | 1.000 | S^{9} | | |
| 0902.0431_FO1718 | 93 | 1.000 | i\left(E_{2}+E_{3}\right) \in S^{9} | | |
| 0902.0431_FO1719 | 93 | 1.000 | \alpha_{23}\left(t_{0}\right) | | |
| 0902.0431_FO1720 | 93 | 1.000 | V^{9}=\left\{X \in V^{10} \mid \tau X=X\right\} | | |
| 0902.0431_FO1721 | 93 | 1.000 | \alpha_{23}\left(t_{0}\right) \in\left(E_{6}\right)_{E_{1}} | | |
| 0902.0431_FO1722 | 93 | 1.000 | \alpha_{23}\left(t_{0}\right) E_{1}= | | |
| 0902.0431_FO1723 | 93 | 0.769 | \operatorname{Spin}(9)=\left(F_{4}\right)_{E_{1}} \subset\left(E_{6}\right)_{E_{1}} | | |
| 0902.0431_FO1724 | 93 | 1.000 | \beta \in \operatorname{Spin}(9) | | |
| 0902.0431_FO1725 | 93 | 1.000 | \alpha_{23}(\pi / 2) \in\left(E_{6}\right)_{E_{1}} | | |
| 0902.0431_FO1726 | 93 | 1.000 | \alpha\left(i\left(E_{2}+E_{3}\right)\right)=i\left(E_{2}+E_{3}\right) | | |
| 0902.0431_FO1727 | 93 | 1.000 | \alpha E=\alpha E_{1}+ | | |
| 0902.0431_FO1728 | 93 | 1.000 | \alpha\left(E_{2}+E_{3}\right)=E_{1}+\left(E_{2}+E_{3}\right)=E | | |
| 0902.0431_FO1729 | 93 | 1.000 | \alpha \in\left(F_{4}\right)_{E_{1}}=\operatorname{Spin}(9) | | |
| 0902.0431_FO1730 | 93 | 0.994 | \left(E_{6}\right)_{E_{1}} / \operatorname{Spin}(9) \simeq S^{9} | | |
| 0902.0431_FO1731 | 93 | 1.000 | \quad\left(E_{6}\right)_{E_{1}} \cong \operatorname{Spin}(10) | | |
| 0902.0431_FO1732 | 93 | 1.000 | p(\alpha)=\alpha \mid V^{10} | | |
| 0902.0431_FO1733 | 93 | 1.000 | p:\left(E_{6}\right)_{E_{1}} \rightarrow S O(10) | | |
| 0902.0431_FO1734 | 93 | 0.991 | p^{\prime}: \operatorname{Spin}(9) \rightarrow S O(9) | | |
| 0902.0431_FO1735 | 94 | 1.000 | \operatorname{Ker} p= | | |
| 0902.0431_FO1736 | 94 | 1.000 | \{1, \sigma\} | | |
| 0902.0431_FO1737 | 94 | 1.000 | i\left(E_{2}+E_{3}\right) | | |
| 0902.0431_FO1738 | 94 | 1.000 | \alpha E_{i}=E_{i} | | |
| 0902.0431_FO1739 | 94 | 1.000 | \alpha \in \operatorname{Ker} p^{\prime} | | |
| 0902.0431_FO1740 | 94 | 1.000 | \operatorname{Spin}(10) | | |
| 0902.0431_FO1741 | 94 | 0.999 | S O(10) | | |
| 0902.0431_FO1742 | 94 | 1.000 | \theta \in C, \theta \neq 0 | | |
| 0902.0431_FO1743 | 94 | 1.000 | \phi(\theta): \mathfrak{J}^{C} \rightarrow \mathfrak{J}^{C} | | |
| 0902.0431_FO1744 | 94 | 1.000 | \alpha_{12}(t), \alpha_{13}(t) | | |
| 0902.0431_FO1745 | 94 | 1.000 | U(1)=\{\theta \in C \mid(\tau \theta) \theta=1\} | | |
| 0902.0431_FO1746 | 94 | 0.936 | \mathfrak{J}^{C}=\mathfrak{E}_{1}^{C} \oplus \mathfrak{J}(2, \mathfrak{C})^{C} \oplus\left(\mathfrak{J}^{C}\right)_{-\sigma} | | |
| 0902.0431_FO1747 | 94 | 1.000 | \phi(\theta) \in U(1) | | |
| 0902.0431_FO1748 | 94 | 0.984 | \beta \in \operatorname{Spin}(10) | | |
| 0902.0431_FO1749 | 94 | 0.500 | \phi(\theta) | | |
| 0902.0431_FO1750 | 94 | 0.500 | \beta: \phi(\theta) \beta=\beta \phi(\theta) | | |
| 0902.0431_FO1751 | 94 | 0.887 | \left(E_{6}\right)^{\sigma} \cong(U(1) \times \operatorname{Spin}(10)) / \boldsymbol{Z}_{4}, \quad \boldsymbol{Z}_{4}=\{(1, \phi(1)),(-1, \phi(-1)) | | |
| 0902.0431_FO1752 | 94 | 0.907 | (i, \phi(-i)),(-i, \phi(i))\} | | |
| 0902.0431_FO1753 | 95 | 0.969 | \varphi: U(1) \times \operatorname{Spin}(10) \rightarrow\left(E_{6}\right)^{\sigma} | | |
| 0902.0431_FO1754 | 95 | 1.000 | \theta \in C | | |
| 0902.0431_FO1755 | 95 | 1.000 | (\tau \theta) \theta=1 | | |
| 0902.0431_FO1756 | 95 | 0.992 | \beta=\phi(\theta)^{-1} \alpha | | |
| 0902.0431_FO1757 | 95 | 0.992 | \beta E_{1}=E_{1} | | |
| 0902.0431_FO1758 | 95 | 1.000 | \alpha=\phi(\theta) \beta=\varphi(\theta, \beta) | | |
| 0902.0431_FO1759 | 95 | 1.000 | (U(1) \times \operatorname{Spin}(10)) / \boldsymbol{Z}_{4} \cong\left(E_{6}\right)^{\sigma} | | |
| 0902.0431_FO1760 | 95 | 1.000 | \gamma: \mathfrak{J}^{C} \rightarrow \mathfrak{J}^{C} | | |
| 0902.0431_FO1761 | 95 | 1.000 | \gamma \in E_{6} | | |
| 0902.0431_FO1762 | 95 | 1.000 | \gamma^{2}=1 | | |
| 0902.0431_FO1763 | 95 | 1.000 | \left(E_{6}\right)^{\gamma} | | |
| 0902.0431_FO1764 | 95 | 1.000 | \mathfrak{J}=\mathfrak{J}(3, \boldsymbol{H}) \oplus \boldsymbol{H}^{3} | | |
| 0902.0431_FO1765 | 95 | 1.000 | \{X \in \mathfrak{J}(3, \boldsymbol{H}) \mid \operatorname{tr}(X)=0\} | | |
| 0902.0431_FO1766 | 95 | 1.000 | \mathfrak{J}_{\boldsymbol{H}} | | |
| 0902.0431_FO1767 | 95 | 1.000 | \left(\mathfrak{J}_{\boldsymbol{H}}\right)_{0} | | |
| 0902.0431_FO1768 | 95 | 1.000 | \boldsymbol{C}=\left\{x+y e_{1} \mid x, y \in \boldsymbol{R}\right\} \subset \mathfrak{C} | | |
| 0902.0431_FO1769 | 95 | 1.000 | a=x+y e_{1} \in \boldsymbol{C} | | |
| 0902.0431_FO1770 | 95 | 1.000 | a^{\prime} | | |
| 0902.0431_FO1771 | 95 | 1.000 | x+y i \in C | | |
| 0902.0431_FO1772 | 95 | 1.000 | k: \boldsymbol{H} \rightarrow M(2, C) | | |
| 0902.0431_FO1773 | 96 | 1.000 | M^{*}={ }^{t} \bar{M}, M \in M(3, \boldsymbol{H}) | | |
| 0902.0431_FO1774 | 96 | 1.000 | k(M N)=k(M) k(N) | | |
| 0902.0431_FO1775 | 96 | 0.874 | \quad \tau^{t}(k(M))=k\left(M^{*}\right) | | |
| 0902.0431_FO1776 | 96 | 0.999 | J(k(M))=(\tau(k(M))) J | | |
| 0902.0431_FO1777 | 96 | 0.999 | k(\boldsymbol{a} M)=k(\boldsymbol{a}) k(M) | | |
| 0902.0431_FO1778 | 96 | 1.000 | k: M(3, \boldsymbol{H}) \rightarrow M(6, C) | | |
| 0902.0431_FO1779 | 96 | 1.000 | k: \boldsymbol{H}^{3} \rightarrow M(2,6, C) | | |
| 0902.0431_FO1780 | 96 | 1.000 | k: M(3, \boldsymbol{H})^{C} \rightarrow M(6, C) | | |
| 0902.0431_FO1781 | 96 | 1.000 | k:\left(\boldsymbol{H}^{3}\right)^{C} \rightarrow M(2,6, C) | | |
| 0902.0431_FO1782 | 96 | 0.539 | (1) \sim(4) | | |
| 0902.0431_FO1783 | 96 | 1.000 | \mathfrak{S}(6, C) | | |
| 0902.0431_FO1784 | 96 | 1.000 | k_{J}: \mathfrak{J}(3, \boldsymbol{H})^{C} \rightarrow \mathfrak{S}(6, C) | | |
| 0902.0431_FO1785 | 96 | 0.999 | k_{J} | | |
| 0902.0431_FO1786 | 96 | 0.999 | M=M_{1}+i M_{2} \in \mathfrak{J}(3, \boldsymbol{H})^{C} | | |
| 0902.0431_FO1787 | 96 | 0.985 | \langle S, T\rangle | | |
| 0902.0431_FO1788 | 96 | 0.985 | \langle P, Q\rangle | | |
| 0902.0431_FO1789 | 96 | 0.985 | M(2,6 | | |
| 0902.0431_FO1790 | 96 | 1.000 | k: M(3, \boldsymbol{H})^{C} \rightarrow M(6, C), k:\left(\boldsymbol{H}^{3}\right)^{C} \rightarrow M(2,6, C) | | |
| 0902.0431_FO1791 | 97 | 1.000 | a^{\prime}=b^{\prime}=c^{\prime}=d^{\prime}=0 | | |
| 0902.0431_FO1792 | 97 | 1.000 | a=b=c=d=0 | | |
| 0902.0431_FO1793 | 97 | 0.998 | \operatorname{dim}_{C}\left(M(3, \boldsymbol{H})^{C}\right)=36=\operatorname{dim}_{C}(M(6, C)) | | |
| 0902.0431_FO1794 | 97 | 1.000 | m_{k}=a_{k}+b_{k} e_{2} | | |
| 0902.0431_FO1795 | 97 | 1.000 | n_{k}=c_{k}+d_{k} e_{2}, a_{k}, b_{k}, c_{k}, d_{k} \in \boldsymbol{C}, k=1,2 | | |
| 0902.0431_FO1796 | 97 | 1.000 | \operatorname{det}\left(k_{J}(M)\right)=\operatorname{det}(k(M)) | | |
| 0902.0431_FO1797 | 97 | 1.000 | \operatorname{det} M \in C | | |
| 0902.0431_FO1798 | 97 | 1.000 | \operatorname{det} S | | |
| 0902.0431_FO1799 | 97 | 1.000 | s_{i j} | | |
| 0902.0431_FO1800 | 97 | 1.000 | k_{J}(M) | | |
| 0902.0431_FO1801 | 97 | 1.000 | E_{6, \boldsymbol{H}} | | |
| 0902.0431_FO1802 | 98 | 0.924 | \operatorname{Sp}(3) / \boldsymbol{Z}_{2} | | |
| 0902.0431_FO1803 | 98 | 0.999 | \mathfrak{e}_{6, \boldsymbol{H}} | | |
| 0902.0431_FO1804 | 98 | 1.000 | \mathfrak{f}_{4, \boldsymbol{H}}=\left\{\delta \in \mathfrak{e}_{6, \boldsymbol{H}} \mid \delta E=0\right\} | | |
| 0902.0431_FO1805 | 98 | 1.000 | \mathfrak{s p}(3) | | |
| 0902.0431_FO1806 | 98 | 1.000 | \varphi_{*}: \mathfrak{s p}(3) \rightarrow \mathfrak{f}_{4, \boldsymbol{H}} | | |
| 0902.0431_FO1807 | 98 | 0.999 | \operatorname{dim} \mathfrak{e}_{6, \boldsymbol{H}}=21+15=35 | | |
| 0902.0431_FO1808 | 98 | 0.913 | \quad E_{6, \boldsymbol{H}} \cong S U(6) / \boldsymbol{Z}_{2}, \quad \boldsymbol{Z}_{2}=\{E,-E\} | | |
| 0902.0431_FO1809 | 98 | 1.000 | S U(6)=\left\{A \in M(6, C) \mid\left(\tau^{t} A\right) A=E, \operatorname{det} A=1\right\} | | |
| 0902.0431_FO1810 | 98 | 0.963 | \varphi: S U(6) \rightarrow E_{6, \boldsymbol{H}} | | |
| 0902.0431_FO1811 | 98 | 1.000 | \varphi(A) \in E_{6, \boldsymbol{H}} | | |
| 0902.0431_FO1812 | 98 | 1.000 | (\operatorname{det}(\varphi(A) M))^{2}= | | |
| 0902.0431_FO1813 | 98 | 0.933 | \operatorname{det}\left(k_{J}(\varphi(A) M)\right)( | | |
| 0902.0431_FO1814 | 98 | 0.933 | )=\operatorname{det}\left(A\left(k_{J}(M)\right)^{t} A\right)=\operatorname{det}\left(k_{J}(M)\right)=(\operatorname{det} M)^{2} | | |
| 0902.0431_FO1815 | 98 | 1.000 | \operatorname{det}(\varphi(A) M)= \pm \operatorname{det} M | | |
| 0902.0431_FO1816 | 98 | 1.000 | \operatorname{det}(\varphi(A) M) | | |
| 0902.0431_FO1817 | 98 | 1.000 | \operatorname{det} M \neq 0 | | |
| 0902.0431_FO1818 | 99 | 0.996 | \operatorname{Ker} \varphi=\{E,-E\}=\boldsymbol{Z}_{2} | | |
| 0902.0431_FO1819 | 99 | 0.999 | \operatorname{dim} S U(6)=35=\operatorname{dim} E_{6, \boldsymbol{H}} | | |
| 0902.0431_FO1820 | 99 | 1.000 | S U(6) / \boldsymbol{Z}_{2} \cong E_{6, \boldsymbol{H}} | | |
| 0902.0431_FO1821 | 99 | 0.470 | \quad\left(E_{6}\right)^{\gamma} \cong(S p(1) \times S U(6)) / \boldsymbol{Z}_{2}, \quad \boldsymbol{Z}_{2}=\{(1, E),(-1,-E)\} | | |
| 0902.0431_FO1822 | 99 | 0.993 | \varphi: \operatorname{Sp}(1) \times \operatorname{SU}(6) \rightarrow\left(E_{6}\right)^{\gamma} | | |
| 0902.0431_FO1823 | 99 | 1.000 | \varphi(p, A) \in\left(E_{6}\right)^{\gamma} | | |
| 0902.0431_FO1824 | 99 | 0.925 | { }^{t} \varphi(p, A)^{-1}=\tau \varphi(p, A) \tau | | |
| 0902.0431_FO1825 | 99 | 1.000 | \tau^{t} \varphi(p, A) \tau=\varphi\left(\bar{p}, \tau^{t} A\right) | | |
| 0902.0431_FO1826 | 99 | 0.833 | \quad \varphi(p, A) \in\left(E_{6}\right)^{\gamma} | | |
| 0902.0431_FO1827 | 99 | 1.000 | \alpha=\varphi(p, A) | | |
| 0902.0431_FO1828 | 99 | 1.000 | \alpha M \times \alpha N={ }^{t} \alpha^{-1}(M \times N) | | |
| 0902.0431_FO1829 | 99 | 1.000 | \operatorname{det}(\alpha M)=\operatorname{det} M | | |
| 0902.0431_FO1830 | 100 | 1.000 | \varphi(p, A) \in E_{6} | | |
| 0902.0431_FO1831 | 100 | 1.000 | \alpha \in\left(E_{6}\right)^{\gamma} | | |
| 0902.0431_FO1832 | 100 | 1.000 | \alpha^{\prime}= | | |
| 0902.0431_FO1833 | 100 | 1.000 | \alpha \mid\left(\mathfrak{J}_{\boldsymbol{H}}\right)^{C} | | |
| 0902.0431_FO1834 | 100 | 1.000 | \left(\mathfrak{J}_{\boldsymbol{H}}\right)^{C} | | |
| 0902.0431_FO1835 | 100 | 1.000 | A \in S U(6) | | |
| 0902.0431_FO1836 | 100 | 1.000 | \alpha^{\prime}=\varphi(A) | | |
| 0902.0431_FO1837 | 100 | 1.000 | \beta \mid\left(\mathfrak{J}_{\boldsymbol{H}}\right)^{C}=1 | | |
| 0902.0431_FO1838 | 100 | 0.972 | p \in S p(1) | | |
| 0902.0431_FO1839 | 100 | 0.822 | (S p(1) \times S U(6)) / \boldsymbol{Z}_{2} \cong\left(E_{6}\right)^{\gamma} | | |
| 0902.0431_FO1840 | 100 | 1.000 | \left(E_{6}\right)^{\tau \gamma} | | |
| 0902.0431_FO1841 | 100 | 1.000 | \left(\mathfrak{J}^{C}\right)_{\tau \gamma} | | |
| 0902.0431_FO1842 | 100 | 1.000 | \left(\mathfrak{J}^{C}\right)_{-\tau \gamma} | | |
| 0902.0431_FO1843 | 101 | 0.993 | \left(\mathfrak{J}^{C}\right)_{\tau \gamma}: \mathfrak{J}^{C}=\left(\left(\mathfrak{J}^{C}\right)_{\tau \gamma}\right)^{C} | | |
| 0902.0431_FO1844 | 101 | 0.997 | P \circ Q | | |
| 0902.0431_FO1845 | 101 | 0.997 | (P, Q) | | |
| 0902.0431_FO1846 | 101 | 0.984 | \operatorname{Sp}(4) | | |
| 0902.0431_FO1847 | 101 | 0.984 | \mathfrak{J}(4, \boldsymbol{H}) | | |
| 0902.0431_FO1848 | 101 | 0.984 | \mu: \operatorname{Sp}(4) \times \mathfrak{J}(4, \boldsymbol{H}) \rightarrow \mathfrak{J}(4, \boldsymbol{H}) | | |
| 0902.0431_FO1849 | 101 | 1.000 | \mu(A, P)=A P A^{*} | | |
| 0902.0431_FO1850 | 101 | 0.999 | \boldsymbol{H} P_{3} | | |
| 0902.0431_FO1851 | 101 | 1.000 | \mathfrak{J}(4, \boldsymbol{H})^{C} | | |
| 0902.0431_FO1852 | 102 | 1.000 | \mathfrak{J}(4, \boldsymbol{H})_{0} | | |
| 0902.0431_FO1853 | 102 | 1.000 | \{P \in \mathfrak{J}(4, \boldsymbol{H}) \mid \operatorname{tr}(P)=0\} | | |
| 0902.0431_FO1854 | 102 | 1.000 | \mathfrak{J}(4, \boldsymbol{H})_{0}{ }^{C} | | |
| 0902.0431_FO1855 | 102 | 1.000 | g: \mathfrak{J}^{C} \rightarrow \mathfrak{J}(4, \boldsymbol{H})_{0}{ }^{C} | | |
| 0902.0431_FO1856 | 102 | 1.000 | g | | |
| 0902.0431_FO1857 | 102 | 1.000 | g:\left(\mathfrak{J}^{C}\right)_{\tau \gamma} \rightarrow \mathfrak{J}(4, \boldsymbol{H})_{0} | | |
| 0902.0431_FO1858 | 102 | 1.000 | \operatorname{dim}_{C} \mathfrak{J}^{C}=27=\operatorname{dim}_{C} \mathfrak{J}(4, \boldsymbol{H})_{0}{ }^{C}, g | | |
| 0902.0431_FO1859 | 102 | 1.000 | X=M+\boldsymbol{a} | | |
| 0902.0431_FO1860 | 102 | 1.000 | Y=N+\boldsymbol{b} \in\left(\mathfrak{J}_{\boldsymbol{H}}\right)^{C} \oplus\left(\boldsymbol{H}^{3}\right)^{C}=\mathfrak{J}^{C} | | |
| 0902.0431_FO1861 | 103 | 1.000 | g X \circ g Y=g(\gamma(X \times Y))+\frac{1}{4}(\gamma X, Y) E | | |
| 0902.0431_FO1862 | 103 | 1.000 | (\gamma X, Y)=\langle X, Y\rangle | | |
| 0902.0431_FO1863 | 103 | 1.000 | X, Y \in\left(\mathfrak{J}^{C}\right)_{\tau \gamma} | | |
| 0902.0431_FO1864 | 103 | 0.952 | \quad\left(E_{6}\right)^{\tau \gamma} \cong \operatorname{Sp}(4) / \boldsymbol{Z}_{2}, \quad \boldsymbol{Z}_{2}=\{E,-E\} | | |
| 0902.0431_FO1865 | 103 | 0.995 | \varphi: \operatorname{Sp}(4) \rightarrow\left(E_{6}\right)^{\tau \gamma} | | |
| 0902.0431_FO1866 | 103 | 1.000 | \varphi(A) \in\left(E_{6}\right)^{\tau \gamma} | | |
| 0902.0431_FO1867 | 103 | 1.000 | Z=\varphi(A) X | | |
| 0902.0431_FO1868 | 103 | 1.000 | \varphi(A) \in E_{6} | | |
| 0902.0431_FO1869 | 103 | 1.000 | \tau \gamma \varphi(A)=\varphi(A) \tau \gamma | | |
| 0902.0431_FO1870 | 103 | 1.000 | \mathfrak{J}^{C}=\left(\left(\mathfrak{J}^{C}\right)_{\tau \gamma}\right)^{C} | | |
| 0902.0431_FO1871 | 103 | 1.000 | X \in\left(\mathfrak{J}^{C}\right)_{\tau \gamma} | | |
| 0902.0431_FO1872 | 103 | 1.000 | g X \in | | |
| 0902.0431_FO1873 | 103 | 1.000 | \varphi(A) X \in\left(\mathfrak{J}^{C}\right)_{\tau \gamma} | | |
| 0902.0431_FO1874 | 103 | 1.000 | \tau \gamma \varphi(A) \tau \gamma X=\tau \gamma \varphi(A) X=\varphi(A) X | | |
| 0902.0431_FO1875 | 103 | 1.000 | \alpha \in\left(E_{6}\right)^{\tau \gamma} | | |
| 0902.0431_FO1876 | 104 | 1.000 | P \in \mathfrak{J}(4, \boldsymbol{H}) | | |
| 0902.0431_FO1877 | 104 | 1.000 | P^{2}=P, \operatorname{tr}(P)=1 | | |
| 0902.0431_FO1878 | 104 | 1.000 | P \in \boldsymbol{H} P^{3} | | |
| 0902.0431_FO1879 | 104 | 0.999 | A \in \operatorname{Sp}(4) | | |
| 0902.0431_FO1880 | 104 | 1.000 | g E=2 E_{1}-\frac{1}{2} E | | |
| 0902.0431_FO1881 | 104 | 1.000 | \beta E=E | | |
| 0902.0431_FO1882 | 104 | 1.000 | \tau \gamma \beta=\beta \tau \gamma | | |
| 0902.0431_FO1883 | 104 | 1.000 | \tau \beta=\beta \tau | | |
| 0902.0431_FO1884 | 104 | 1.000 | \gamma \beta=\beta \gamma | | |
| 0902.0431_FO1885 | 104 | 0.979 | \beta \in\left(F_{4}\right)^{\gamma} | | |
| 0902.0431_FO1886 | 104 | 0.979 | D \in \operatorname{Sp}(3) | | |
| 0902.0431_FO1887 | 104 | 0.911 | B=\left(\begin{array}{cc}p & 0 \\ 0 & D\end{array}\right) | | |
| 0902.0431_FO1888 | 104 | 0.911 | B \in \operatorname{Sp}(4) | | |
| 0902.0431_FO1889 | 104 | 1.000 | M+\boldsymbol{a} \in\left(\mathfrak{J}_{\boldsymbol{H}}\right)^{C} \oplus\left(\boldsymbol{H}^{3}\right)^{C}=\mathfrak{J}^{C} | | |
| 0902.0431_FO1890 | 105 | 0.806 | \operatorname{Sp}(4) / \boldsymbol{Z}_{2} \cong\left(E_{6}\right)^{\tau \gamma} | | |
| 0902.0431_FO1891 | 105 | 1.000 | (S U(3) \times S U(3) \times S U(3)) / \boldsymbol{Z}_{3} | | |
| 0902.0431_FO1892 | 105 | 1.000 | w: \mathfrak{J}^{C} \rightarrow \mathfrak{J}^{C} | | |
| 0902.0431_FO1893 | 105 | 1.000 | w \in E_{6} | | |
| 0902.0431_FO1894 | 105 | 1.000 | \left(E_{6}\right)^{w} | | |
| 0902.0431_FO1895 | 105 | 1.000 | \{X \in \mathfrak{J}(3, \boldsymbol{C}) \mid \operatorname{tr}(X)=0\} | | |
| 0902.0431_FO1896 | 105 | 1.000 | \mathfrak{J}_{\boldsymbol{C}} | | |
| 0902.0431_FO1897 | 105 | 1.000 | \left(\mathfrak{J}_{\boldsymbol{C}}\right)_{0} | | |
| 0902.0431_FO1898 | 105 | 1.000 | E_{6, \boldsymbol{C}} | | |
| 0902.0431_FO1899 | 105 | 1.000 | \left(\mathfrak{J}_{\boldsymbol{C}}\right)^{C} | | |
| 0902.0431_FO1900 | 105 | 1.000 | F_{4, \boldsymbol{C}}=\left\{\alpha \in \operatorname{Iso}_{\boldsymbol{R}}\left(\mathfrak{J}_{\boldsymbol{C}}\right) \mid \alpha(X \circ Y)=\alpha X \circ \alpha Y\right\} | | |
| 0902.0431_FO1901 | 105 | 0.998 | \left(S U(3) / \boldsymbol{Z}_{3}\right) \cdot \boldsymbol{Z}_{2} | | |
| 0902.0431_FO1902 | 105 | 1.000 | \mathfrak{e}_{6, \boldsymbol{C}} | | |
| 0902.0431_FO1903 | 105 | 1.000 | \mathfrak{f}_{4, \boldsymbol{C}}=\left\{\delta \in \mathfrak{e}_{6, \boldsymbol{C}} \mid \delta E=0\right\} | | |
| 0902.0431_FO1904 | 106 | 1.000 | \pi_{0}\left(F_{4, \boldsymbol{C}}\right) \rightarrow \pi_{0}\left(E_{6, \boldsymbol{C}}\right) \rightarrow \pi_{0}\left(E I V_{\boldsymbol{C}}\right) | | |
| 0902.0431_FO1905 | 106 | 1.000 | \boldsymbol{Z}_{2} \rightarrow \pi_{0}\left(E_{6, \boldsymbol{C}}\right) \rightarrow 0 | | |
| 0902.0431_FO1906 | 106 | 1.000 | \pi_{0}\left(E_{6, \boldsymbol{C}}\right) | | |
| 0902.0431_FO1907 | 106 | 1.000 | h: \boldsymbol{C} \oplus \boldsymbol{C} \rightarrow \boldsymbol{C}^{C} | | |
| 0902.0431_FO1908 | 106 | 1.000 | h: M(3, \boldsymbol{C}) \oplus M(3, \boldsymbol{C}) \rightarrow M(3, \boldsymbol{C})^{C} | | |
| 0902.0431_FO1909 | 106 | 1.000 | h: M(3, \boldsymbol{C}) \oplus M(3, \boldsymbol{C}) \rightarrow | | |
| 0902.0431_FO1910 | 106 | 1.000 | M(3, \boldsymbol{C})^{C} | | |
| 0902.0431_FO1911 | 106 | 0.999 | h(a, b) h\left(a^{\prime}, b^{\prime}\right)=h\left(a a^{\prime}, b b^{\prime}\right), h(A, B) h\left(A^{\prime}, B^{\prime}\right)=h\left(A A^{\prime}, B B^{\prime}\right) | | |
| 0902.0431_FO1912 | 106 | 0.989 | \tau h(a, b)=h(b, a), \overline{h(a, b)}=h(\bar{b}, \bar{a}) | | |
| 0902.0431_FO1913 | 106 | 1.000 | \operatorname{det}(h(A, B))=h(\operatorname{det} A, \operatorname{det} B) | | |
| 0902.0431_FO1914 | 106 | 1.000 | \iota^{2}=\iota, \bar{\iota}^{2}=\bar{\iota}, \iota+\bar{\iota}=1 | | |
| 0902.0431_FO1915 | 106 | 0.956 | \quad \mathfrak{e}_{6, C} \cong \mathfrak{s u}(3) \oplus \mathfrak{s u}(3) | | |
| 0902.0431_FO1916 | 106 | 1.000 | \phi_{\boldsymbol{C}}: \mathfrak{s u}(3) \oplus \mathfrak{s u}(3) \rightarrow \mathfrak{e}_{6, \boldsymbol{C}} | | |
| 0902.0431_FO1917 | 106 | 0.878 | \operatorname{SU}(3) \times \operatorname{SU}(3) | | |
| 0902.0431_FO1918 | 106 | 0.990 | (S U(3) \times S U(3)) \cdot \boldsymbol{Z}_{2} | | |
| 0902.0431_FO1919 | 106 | 0.990 | S U(3) \times S U(3) | | |
| 0902.0431_FO1920 | 107 | 0.874 | E_{6, \boldsymbol{C}} \cong\left((S U(3) \times S U(3)) / \boldsymbol{Z}_{3}\right) \cdot \boldsymbol{Z}_{2}, \boldsymbol{Z}_{3}=\left\{(E, E),\left(\omega_{1} E\right.\right. | | |
| 0902.0431_FO1921 | 107 | 0.763 | \left.\left.\omega_{1} E\right),\left(\omega_{1}{ }^{2} E, \omega_{1}{ }^{2} E\right)\right\}, \omega_{1}=-\frac{1}{2}+\frac{\sqrt{3}}{2} e_{1} | | |
| 0902.0431_FO1922 | 107 | 1.000 | \varphi:(S U(3) \times S U(3)) \cdot \boldsymbol{Z}_{2} \rightarrow E_{6, \boldsymbol{C}} | | |
| 0902.0431_FO1923 | 107 | 0.975 | \alpha=\varphi((A, B), 1) \in E_{6, \boldsymbol{C}} | | |
| 0902.0431_FO1924 | 107 | 0.975 | \operatorname{det}(h(A, B))=h(\operatorname{det} A | | |
| 0902.0431_FO1925 | 107 | 0.862 | \operatorname{det} B)(\operatorname{Lemma} 3.13 .2 .(4))=h(1,1)=1 | | |
| 0902.0431_FO1926 | 107 | 0.862 | \tau h(A, B)^{*} h(A, B)=h\left(A^{*}, B^{*}\right) h(A, B) | | |
| 0902.0431_FO1927 | 107 | 0.963 | =h\left(A^{*} A, B^{*} B\right)=h(E, E)=E | | |
| 0902.0431_FO1928 | 107 | 1.000 | \alpha \in E_{6, \boldsymbol{C}} | | |
| 0902.0431_FO1929 | 107 | 1.000 | \varphi((E, E), \epsilon)=\epsilon \in G_{2, \boldsymbol{C}}(=\operatorname{Aut}(\boldsymbol{C})) \subset F_{4, \boldsymbol{C}} \subset E_{6, \boldsymbol{C}} | | |
| 0902.0431_FO1930 | 107 | 0.997 | \varphi((A, B), \epsilon)=\varphi((A, B), 1) \varphi((E, E), \epsilon) \in E_{6, \boldsymbol{C}} | | |
| 0902.0431_FO1931 | 107 | 1.000 | \operatorname{Ker} \varphi=\left\{(E, E),\left(\omega_{1} E, \omega_{1} E\right),\left(\omega_{1}{ }^{2} E, \omega_{1}{ }^{2} E\right)\right\} \times 1=\boldsymbol{Z}_{3} \times 1 | | |
| 0902.0431_FO1932 | 107 | 1.000 | \operatorname{Ker} \varphi | | |
| 0902.0431_FO1933 | 107 | 0.725 | \operatorname{dim}(\mathfrak{s u}(3) \oplus \mathfrak{s u}(3))=\operatorname{dim}\left(\mathfrak{e}_{6, \boldsymbol{C}}\right) | | |
| 0902.0431_FO1934 | 107 | 0.725 | \left(\varphi_{*}\right. | | |
| 0902.0431_FO1935 | 107 | 0.924 | \phi_{\boldsymbol{C}} | | |
| 0902.0431_FO1936 | 107 | 0.924 | \varphi: S U(3) \times S U(3) \rightarrow | | |
| 0902.0431_FO1937 | 107 | 1.000 | \left(E_{6, \boldsymbol{C}}\right)_{0} | | |
| 0902.0431_FO1938 | 107 | 0.993 | \epsilon=\varphi((E, E), \epsilon) \notin\left(E_{6, \boldsymbol{C}}\right)_{0} | | |
| 0902.0431_FO1939 | 107 | 0.993 | A, B \in S U(3) | | |
| 0902.0431_FO1940 | 107 | 1.000 | \left((S U(3) \times S U(3)) / \boldsymbol{Z}_{3}\right) \cdot \boldsymbol{Z}_{2} \cong E_{6, \boldsymbol{C}} | | |
| 0902.0431_FO1941 | 108 | 0.588 | \left(E_{6}\right)^{w} \cong(S U(3) \times S U(3) \times S U(3)) / \boldsymbol{Z}_{3}, \quad \boldsymbol{Z}_{3}=\{(E, E, E) | | |
| 0902.0431_FO1942 | 108 | 0.929 | \left.\left(\omega_{1} E, \omega_{1} E, \omega_{1} E\right),\left(\omega_{1}^{2} E, \omega_{1}^{2} E, \omega_{1}^{2} E\right)\right\}, \omega_{1}=-\frac{1}{2}+\frac{\sqrt{3}}{2} e_{1} | | |
| 0902.0431_FO1943 | 108 | 0.917 | \varphi: S U(3) \times S U(3) \times S U(3) \rightarrow\left(E_{6}\right)^{w} | | |
| 0902.0431_FO1944 | 108 | 1.000 | \alpha=\varphi(P, A, B) \in\left(E_{6}\right)^{w} | | |
| 0902.0431_FO1945 | 108 | 1.000 | X+M, Y+N \in\left(\mathfrak{J}_{\boldsymbol{C}}\right)^{C} \oplus M(3, \boldsymbol{C})^{C}=\mathfrak{J}^{C} | | |
| 0902.0431_FO1946 | 108 | 1.000 | w \alpha=\alpha w | | |
| 0902.0431_FO1947 | 108 | 1.000 | \alpha \in\left(E_{6}\right)^{w} | | |
| 0902.0431_FO1948 | 108 | 1.000 | \left(\mathfrak{J}^{C}\right)_{w}=\left\{X \in \mathfrak{J}^{C} \mid w X=X\right\}=\left(\mathfrak{J}_{\boldsymbol{C}}\right)^{C} | | |
| 0902.0431_FO1949 | 108 | 1.000 | E_{6, \boldsymbol{C}}: \alpha^{\prime} \in E_{6, \boldsymbol{C}} | | |
| 0902.0431_FO1950 | 108 | 1.000 | \beta=\varphi(E, A, B)^{-1} \alpha | | |
| 0902.0431_FO1951 | 108 | 1.000 | \beta \mid\left(\mathfrak{J}_{\boldsymbol{C}}\right)^{C}=1 | | |
| 0902.0431_FO1952 | 109 | 1.000 | \gamma_{1}: \mathfrak{J}^{C} \rightarrow \mathfrak{J}^{C}, \gamma_{1}(X+M)=\bar{X}+\bar{M}, X+ | | |
| 0902.0431_FO1953 | 109 | 1.000 | M \in\left(\mathfrak{J}_{\boldsymbol{C}}\right)^{C} \oplus M(3, \boldsymbol{C})^{C}=\mathfrak{J}^{C} | | |
| 0902.0431_FO1954 | 109 | 1.000 | \gamma_{1} \in G_{2} \subset F_{4} \subset E_{6} | | |
| 0902.0431_FO1955 | 109 | 1.000 | \beta=\alpha^{-1} \varphi(E, A, B) \gamma_{1} | | |
| 0902.0431_FO1956 | 109 | 1.000 | \beta \in E_{6} | | |
| 0902.0431_FO1957 | 109 | 1.000 | \subset\left(E_{6}\right)^{w} | | |
| 0902.0431_FO1958 | 109 | 1.000 | \varphi(E, A, B) \in\left(E_{6}\right)^{w} | | |
| 0902.0431_FO1959 | 109 | 1.000 | \gamma_{1} \in\left(E_{6}\right)^{w} | | |
| 0902.0431_FO1960 | 109 | 0.837 | \operatorname{Ker} \varphi=\left\{(E, E, E),\left(\omega_{1} E, \omega_{1} E, \omega_{1} E\right),\left(\omega_{1}{ }^{2} E, \omega_{1}{ }^{2} E, \omega_{1}{ }^{2} E\right)\right\}=\boldsymbol{Z}_{3} | | |
| 0902.0431_FO1961 | 109 | 0.996 | (S U(3) \times S U(3) \times S U(3)) / \boldsymbol{Z}_{3} \cong | | |
| 0902.0431_FO1962 | 109 | 0.998 | S U(3) \times S U(3) \times S U(3) \rightarrow\left(E_{6}\right)^{w} | | |
| 0902.0431_FO1963 | 109 | 0.997 | \left(\mathfrak{e}_{6}\right)^{w} | | |
| 0902.0431_FO1964 | 109 | 0.997 | \operatorname{dim}\left(\left(\mathfrak{e}_{6}\right)^{w}\right)=16+8=24=8+8+8=\operatorname{dim}(\mathfrak{s u}(3) \oplus | | |
| 0902.0431_FO1965 | 109 | 0.971 | \mathfrak{s} \mathfrak{u}(3) \oplus \mathfrak{s} \mathfrak{u}(3)) | | |
| 0902.0431_FO1966 | 109 | 1.000 | \left((S U(3) \times S U(3) \times S U(3)) / \boldsymbol{Z}_{3}\right) \cdot \boldsymbol{Z}_{2} | | |
| 0902.0431_FO1967 | 109 | 1.000 | S U(3) \times S U(3) \times S U(3) | | |
| 0902.0431_FO1968 | 109 | 1.000 | \left.\gamma_{1}(P, A, B)=(\bar{P}, \bar{B}, \bar{A})\right) | | |
| 0902.0431_FO1969 | 109 | 0.999 | \alpha^{*} | | |
| 0902.0431_FO1970 | 109 | 0.999 | \alpha \in E_{6}{ }^{C} | | |
| 0902.0431_FO1971 | 109 | 1.000 | \alpha^{*}=\tau^{t} \alpha \tau \in E_{6}{ }^{C} | | |
| 0902.0431_FO1972 | 110 | 1.000 | \langle X, Y\rangle_{\gamma} | | |
| 0902.0431_FO1973 | 110 | 1.000 | \langle X, Y\rangle_{\sigma} | | |
| 0902.0431_FO1974 | 110 | 1.000 | \mathfrak{J}\left(3, \mathfrak{C}^{C}\right) | | |
| 0902.0431_FO1975 | 111 | 0.906 | \left(\begin{array}{l}X \\ Y \\ \xi \\ \eta\end{array}\right) | | |
| 0902.0431_FO1976 | 111 | 0.996 | (X, Y, \xi, \eta) | | |
| 0902.0431_FO1977 | 111 | 0.996 | \dot{X}+Y+\dot{\xi}+\eta | | |
| 0902.0431_FO1978 | 111 | 1.000 | \{P, Q\} | | |
| 0902.0431_FO1979 | 111 | 1.000 | P=(X, Y, \xi, \eta), Q=(Z, W, \zeta, \omega) \in \mathfrak{P}^{C} | | |
| 0902.0431_FO1980 | 111 | 0.977 | \phi \in \mathfrak{e}_{6}{ }^{C}, A, B \in \mathfrak{J}^{C}, \nu \in C | | |
| 0902.0431_FO1981 | 111 | 0.977 | \Phi(\phi, A, B, \nu) | | |
| 0902.0431_FO1982 | 111 | 1.000 | \mathfrak{P}^{C} \rightarrow \mathfrak{P}^{C} | | |
| 0902.0431_FO1983 | 111 | 1.000 | P \times Q: \mathfrak{P}^{C} \rightarrow \mathfrak{P}^{C} | | |
| 0902.0431_FO1984 | 112 | 1.000 | P, Q, R \in \mathfrak{P}^{C} | | |
| 0902.0431_FO1985 | 112 | 1.000 | P \times Q=Q \times P | | |
| 0902.0431_FO1986 | 112 | 1.000 | (P \times Q) P-(P \times P) Q+\frac{3}{8}\{P, Q\} P=0 | | |
| 0902.0431_FO1987 | 112 | 1.000 | (P \times R) Q-(Q \times R) P+\frac{1}{8}\{Q, R\} P-\frac{1}{8}\{P, R\} Q-\frac{1}{4}\{P, Q\} R=0 | | |
| 0902.0431_FO1988 | 112 | 0.910 | =\cdots | | |
| 0902.0431_FO1989 | 112 | 0.910 | X \vee Y | | |
| 0902.0431_FO1990 | 112 | 1.000 | P+R | | |
| 0902.0431_FO1991 | 112 | 1.000 | Q | | |
| 0902.0431_FO1992 | 112 | 0.786 | ((\mathrm{i})-(\mathrm{ii})) \div 3 | | |
| 0902.0431_FO1993 | 112 | 1.000 | \mathfrak{M}^{C} | | |
| 0902.0431_FO1994 | 112 | 1.000 | \xi \neq 0, \eta \neq 0 | | |
| 0902.0431_FO1995 | 113 | 1.000 | X \vee(X \times X)=0 | | |
| 0902.0431_FO1996 | 113 | 1.000 | (X \times X) \times | | |
| 0902.0431_FO1997 | 113 | 1.000 | (X \times X)=(\operatorname{det} X) X | | |
| 0902.0431_FO1998 | 113 | 0.999 | \alpha \in E_{7}{ }^{C} | | |
| 0902.0431_FO1999 | 113 | 0.999 | \alpha \in E_{7} | | |
| 0902.0431_FO2000 | 113 | 1.000 | \alpha \mathfrak{M}^{C} \subset \mathfrak{M}^{C} | | |
| 0902.0431_FO2001 | 113 | 1.000 | P \in \mathfrak{M}^{C} | | |
| 0902.0431_FO2002 | 113 | 1.000 | \alpha P \times \alpha P=\alpha(P \times P) \alpha^{-1}=\alpha 0 \alpha^{-1}=0 | | |
| 0902.0431_FO2003 | 113 | 1.000 | \alpha P \in \mathfrak{M}^{C} | | |
| 0902.0431_FO2004 | 113 | 1.000 | \{\alpha P, \alpha Q\}=\{P, Q\} | | |
| 0902.0431_FO2005 | 113 | 0.974 | \left[\Phi_{1}, \Phi_{2}\right] | | |
| 0902.0431_FO2006 | 114 | 1.000 | \Phi_{i} \in \mathfrak{e}_{7}{ }^{C} | | |
| 0902.0431_FO2007 | 114 | 1.000 | P \in \mathfrak{P}^{C} | | |
| 0902.0431_FO2008 | 114 | 1.000 | \Phi | | |
| 0902.0431_FO2009 | 114 | 0.994 | \Phi \in \operatorname{Hom}_{C}\left(\mathfrak{P}^{C}\right) | | |
| 0902.0431_FO2010 | 114 | 0.994 | \Phi \in \mathfrak{e}_{7}{ }^{C} | | |
| 0902.0431_FO2011 | 114 | 0.994 | \mathfrak{P}^{C}=\mathfrak{J}^{C} \oplus \mathfrak{J}^{C} \oplus C \oplus C, \Phi | | |
| 0902.0431_FO2012 | 114 | 1.000 | 0 \neq r \in C | | |
| 0902.0431_FO2013 | 114 | 1.000 | f_{r}: \mathfrak{P}^{C} \rightarrow \mathfrak{P}^{C} | | |
| 0902.0431_FO2014 | 114 | 1.000 | f_{r} | | |
| 0902.0431_FO2015 | 114 | 0.985 | f_{r} \alpha f_{r}{ }^{-1} \in E_{7}{ }^{C} | | |
| 0902.0431_FO2016 | 115 | 1.000 | \Phi_{-3}(0,0,1,0) \times(0,0,1,0)=0 | | |
| 0902.0431_FO2017 | 115 | 1.000 | \kappa=0 | | |
| 0902.0431_FO2018 | 115 | 1.000 | \Phi_{-3}=0 | | |
| 0902.0431_FO2019 | 115 | 1.000 | \Phi_{3}=0 | | |
| 0902.0431_FO2020 | 115 | 1.000 | \Phi_{-2}(0,0,1,0) \times(0,0,1,0)=0 | | |
| 0902.0431_FO2021 | 115 | 1.000 | C=0 | | |
| 0902.0431_FO2022 | 115 | 0.997 | \Phi_{-2} P \times P=0 | | |
| 0902.0431_FO2023 | 115 | 0.997 | P=(Y \times Y, Y, 1, \operatorname{det} Y) \in \mathfrak{M}^{C} | | |
| 0902.0431_FO2024 | 115 | 0.997 | (0,0,0, d(Y)) \times | | |
| 0902.0431_FO2025 | 115 | 1.000 | (Y \times Y, Y, 1, \operatorname{det} Y)=0 | | |
| 0902.0431_FO2026 | 115 | 1.000 | d=0 | | |
| 0902.0431_FO2027 | 115 | 1.000 | \Phi_{-2}=0 | | |
| 0902.0431_FO2028 | 115 | 1.000 | \Phi_{2}=0 | | |
| 0902.0431_FO2029 | 115 | 0.997 | \Phi_{-1} P \times P=0 | | |
| 0902.0431_FO2030 | 115 | 0.997 | (l(Y), B, 0, b(Y \times Y)) \times | | |
| 0902.0431_FO2031 | 115 | 1.000 | P=(X, X \times X, \operatorname{det} X, 1) \in \mathfrak{M}^{C} | | |
| 0902.0431_FO2032 | 115 | 0.942 | (l(X \times X),(\operatorname{det} X) B, 0, b(X)) \times(X, X \times X, \operatorname{det} X, 1)=0 | | |
| 0902.0431_FO2033 | 115 | 1.000 | 3(\operatorname{det} X)(B, X)=3(\operatorname{det} X) b(X) | | |
| 0902.0431_FO2034 | 115 | 1.000 | \Phi_{1} | | |
| 0902.0431_FO2035 | 116 | 1.000 | \Phi_{0} P \times P=0 | | |
| 0902.0431_FO2036 | 116 | 1.000 | (g(X), h(X \times | | |
| 0902.0431_FO2037 | 116 | 0.998 | X),(\operatorname{det} X) \nu, \mu) \times(X, X \times X, \operatorname{det} X, 1)=0 | | |
| 0902.0431_FO2038 | 116 | 0.983 | \phi=g-\frac{1}{3}(\nu+2 \mu) 1 | | |
| 0902.0431_FO2039 | 116 | 1.000 | \psi=h-\frac{1}{3}(2 \nu+\mu) 1 \in \mathfrak{e}_{6}{ }^{C} | | |
| 0902.0431_FO2040 | 116 | 0.999 | 2 \psi(X \times X) \times(X \times X)=(\operatorname{det} X) \phi X | | |
| 0902.0431_FO2041 | 116 | 0.999 | -{ }^{t} \psi | | |
| 0902.0431_FO2042 | 116 | 0.999 | \psi^{\prime} | | |
| 0902.0431_FO2043 | 116 | 1.000 | \psi^{\prime}((X \times X) \times(X \times X))=(\operatorname{det} X) \phi X | | |
| 0902.0431_FO2044 | 116 | 1.000 | (\operatorname{det} X) \psi^{\prime} X= | | |
| 0902.0431_FO2045 | 116 | 1.000 | (\operatorname{det} X) \phi X | | |
| 0902.0431_FO2046 | 116 | 1.000 | \psi^{\prime} X=\phi X, X \in \mathfrak{J}^{C} | | |
| 0902.0431_FO2047 | 116 | 1.000 | \{\Phi(0,0,1,0),(0,0,0,1)\}+\{(0,0,1,0), \Phi(0,0,0,1)\}=0 | | |
| 0902.0431_FO2048 | 116 | 0.990 | \nu+\mu=0 | | |
| 0902.0431_FO2049 | 116 | 0.731 | \Phi=\Phi(\phi, A, B, \nu), \phi \in \mathfrak{e}_{6}{ }^{C}, A, B \in \mathfrak{J}^{C}, \nu \in C | | |
| 0902.0431_FO2050 | 116 | 1.000 | \exp t \Phi \in E_{7}{ }^{C} | | |
| 0902.0431_FO2051 | 116 | 1.000 | t \in C | | |
| 0902.0431_FO2052 | 117 | 1.000 | P, Q \in \mathfrak{P}^{C} | | |
| 0902.0431_FO2053 | 117 | 1.000 | [\Phi, P \times P]=2 \Phi P \times P | | |
| 0902.0431_FO2054 | 117 | 1.000 | P=(X, Y, \xi, \eta) \in | | |
| 0902.0431_FO2055 | 117 | 0.783 | \Phi=\Phi(\phi, A, B, \nu), \phi \in \mathfrak{e}_{6}{ }^{C} | | |
| 0902.0431_FO2056 | 117 | 1.000 | A, B \in \mathfrak{J}^{C}, \nu \in C | | |
| 0902.0431_FO2057 | 117 | 1.000 | \alpha \in \operatorname{Hom}_{C}\left(\mathfrak{P}^{C}\right) | | |
| 0902.0431_FO2058 | 117 | 1.000 | (P, Q):\left({ }^{t} \alpha P, Q\right)=(P, \alpha Q) | | |
| 0902.0431_FO2059 | 117 | 1.000 | \lambda \in E_{7} | | |
| 0902.0431_FO2060 | 117 | 1.000 | \lambda^{2}=-1 | | |
| 0902.0431_FO2061 | 118 | 0.999 | \Phi(\phi, A, B, \nu) \in \mathfrak{e}_{7}{ }^{C} | | |
| 0902.0431_FO2062 | 118 | 1.000 | (P, \lambda Q)=\{P, Q\}=\{\alpha P, \alpha Q\} | | |
| 0902.0431_FO2063 | 118 | 1.000 | =(\alpha P, \lambda \alpha Q) | | |
| 0902.0431_FO2064 | 118 | 0.986 | =\left(P,{ }^{t} \alpha \lambda \alpha Q\right) | | |
| 0902.0431_FO2065 | 118 | 0.986 | \lambda={ }^{t} \alpha \lambda \alpha | | |
| 0902.0431_FO2066 | 118 | 0.986 | { }^{t} \alpha^{-1}=\lambda \alpha \lambda^{-1} | | |
| 0902.0431_FO2067 | 118 | 1.000 | \tau \lambda \alpha=\alpha \tau \lambda | | |
| 0902.0431_FO2068 | 118 | 1.000 | \langle\alpha P, \alpha Q\rangle=\{\tau \lambda \alpha P, \alpha Q\}=\{\alpha \tau \lambda P, \alpha Q\} | | |
| 0902.0431_FO2069 | 118 | 1.000 | =\{\tau \lambda P, Q\} | | |
| 0902.0431_FO2070 | 118 | 1.000 | =\langle P, Q\rangle | | |
| 0902.0431_FO2071 | 119 | 1.000 | \tau \Phi(\phi, A | | |
| 0902.0431_FO2072 | 119 | 0.817 | B, \nu) \tau=\Phi(\tau \phi \tau, \tau A, \tau B, \tau \nu) | | |
| 0902.0431_FO2073 | 119 | 0.998 | \Phi^{*} | | |
| 0902.0431_FO2074 | 119 | 0.993 | \Phi^{*}=\tau \lambda \Phi \lambda \tau \in \mathfrak{e}_{7}{ }^{C} | | |
| 0902.0431_FO2075 | 119 | 0.993 | \Phi \in \mathfrak{e}_{7}{ }^{C}, \Phi | | |
| 0902.0431_FO2076 | 119 | 0.822 | \Phi^{*}=-\Phi | | |
| 0902.0431_FO2077 | 119 | 0.999 | \mathfrak{e}_{6}{ }^{C}=\left\{\Phi(\phi, 0,0,0) \in \mathfrak{e}_{7}{ }^{C} \mid \phi \in \mathfrak{e}_{6}{ }^{C}\right\} | | |
| 0902.0431_FO2078 | 119 | 0.999 | \mathfrak{N}^{C}=\left\{\Phi(0, A, B, \nu) \in \mathfrak{e}_{7}{ }^{C} \mid A, B \in\right. | | |
| 0902.0431_FO2079 | 119 | 1.000 | \left.\mathfrak{J}^{C}, \nu \in C\right\} | | |
| 0902.0431_FO2080 | 119 | 1.000 | p: \mathfrak{e}_{7}{ }^{C} \rightarrow \mathfrak{e}_{6}{ }^{C} | | |
| 0902.0431_FO2081 | 119 | 1.000 | q: \mathfrak{e}_{7}{ }^{C} \rightarrow \mathfrak{N}^{C} | | |
| 0902.0431_FO2082 | 119 | 1.000 | \mathfrak{e}_{7}{ }^{C}=\mathfrak{e}_{6}{ }^{C} \oplus \mathfrak{N}^{C} | | |
| 0902.0431_FO2083 | 119 | 1.000 | \phi \in p(\mathfrak{a}) | | |
| 0902.0431_FO2084 | 119 | 1.000 | \Phi(0, A, B, \nu) \in \mathfrak{N}^{C} | | |
| 0902.0431_FO2085 | 119 | 1.000 | \Phi(\phi, A, B, \nu) \in \mathfrak{a} | | |
| 0902.0431_FO2086 | 119 | 1.000 | \phi_{1} \in \mathfrak{e}_{6}{ }^{C} | | |
| 0902.0431_FO2087 | 119 | 1.000 | \left[\phi_{1}, \phi\right] \in p(\mathfrak{a}) | | |
| 0902.0431_FO2088 | 119 | 1.000 | \mathfrak{e}_{6}{ }^{C} \cap \mathfrak{a} \neq\{0\} | | |
| 0902.0431_FO2089 | 119 | 1.000 | \mathfrak{N}^{C} \cap \mathfrak{a} \neq\{0\} | | |
| 0902.0431_FO2090 | 119 | 1.000 | \mathfrak{e}_{6}{ }^{C} \cap \mathfrak{a}= | | |
| 0902.0431_FO2091 | 119 | 1.000 | \{0\} | | |
| 0902.0431_FO2092 | 119 | 1.000 | \mathfrak{N}^{C} \cap \mathfrak{a}=\{0\} | | |
| 0902.0431_FO2093 | 119 | 1.000 | p \mid \mathfrak{a}: \mathfrak{a} \rightarrow \mathfrak{e}_{6}{ }^{C} | | |
| 0902.0431_FO2094 | 119 | 1.000 | \mathfrak{N}^{C} \cap \mathfrak{a}= | | |
| 0902.0431_FO2095 | 119 | 0.999 | p(\mathfrak{a})=\mathfrak{e}_{6}{ }^{C} | | |
| 0902.0431_FO2096 | 119 | 0.996 | \operatorname{dim}_{C}(\mathfrak{a})=\operatorname{dim}_{C}(p(\mathfrak{a}))=\operatorname{dim}_{C}\left(\mathfrak{e}_{6}{ }^{C}\right)=78 | | |
| 0902.0431_FO2097 | 119 | 0.996 | \mathfrak{e}_{6}{ }^{C} \cap \mathfrak{a}=\{0\} | | |
| 0902.0431_FO2098 | 119 | 1.000 | q \mid \mathfrak{a}: \mathfrak{a} \rightarrow \mathfrak{N}^{C} | | |
| 0902.0431_FO2099 | 119 | 1.000 | \operatorname{dim}_{C}(\mathfrak{a}) \leq \operatorname{dim}_{C}\left(\mathfrak{N}^{C}\right)=27+27+1=55 | | |
| 0902.0431_FO2100 | 119 | 1.000 | \mathfrak{e}_{6}{ }^{C} \cap \mathfrak{a}=\mathfrak{e}_{6}{ }^{C} | | |
| 0902.0431_FO2101 | 119 | 1.000 | \mathfrak{a} \supset \mathfrak{e}_{6}{ }^{C} | | |
| 0902.0431_FO2102 | 119 | 1.000 | \Phi\left(0,0, \mathfrak{J}^{C}, 0\right) \subset \mathfrak{a} | | |
| 0902.0431_FO2103 | 120 | 0.602 | \Phi(0,0,0,1) \in \mathfrak{a} | | |
| 0902.0431_FO2104 | 120 | 0.602 | \mathfrak{a} \supset \mathfrak{N}^{C} | | |
| 0902.0431_FO2105 | 120 | 0.602 | \mathfrak{a} \supset \mathfrak{e}_{6}{ }^{C} \oplus \mathfrak{N}^{C}=\mathfrak{e}_{7}{ }^{C} | | |
| 0902.0431_FO2106 | 120 | 1.000 | \Phi(0, A, B, \nu) | | |
| 0902.0431_FO2107 | 120 | 1.000 | \mathfrak{N}^{C} \cap \mathfrak{a} | | |
| 0902.0431_FO2108 | 120 | 1.000 | \Phi(0, A, B, \nu), A \neq 0 | | |
| 0902.0431_FO2109 | 120 | 1.000 | B_{1} \in \mathfrak{J}^{C} | | |
| 0902.0431_FO2110 | 120 | 1.000 | A \vee B_{1} \neq 0 | | |
| 0902.0431_FO2111 | 120 | 1.000 | \left[A \vee B_{1}, \phi\right] \neq 0 | | |
| 0902.0431_FO2112 | 120 | 1.000 | \Phi(0, A, B, \nu), B \neq 0 | | |
| 0902.0431_FO2113 | 120 | 1.000 | \Phi(0,0,0, \nu), \nu \neq 0 | | |
| 0902.0431_FO2114 | 120 | 1.000 | 0 \neq A \in \mathfrak{J}^{C} | | |
| 0902.0431_FO2115 | 120 | 1.000 | \mathfrak{a}=\mathfrak{e}_{7}{ }^{C} | | |
| 0902.0431_FO2116 | 120 | 0.990 | \mathfrak{e}_{7}{ }^{C} \mathfrak{P}^{C}=\left\{\sum_{k} \Phi_{k} P_{k} \mid \Phi_{k} \in \mathfrak{e}_{7}{ }^{C}, P_{k} \in \mathfrak{P}^{C}\right\}=\mathfrak{P}^{C} | | |
| 0902.0431_FO2117 | 120 | 1.000 | (0,0,0,1) \in W | | |
| 0902.0431_FO2118 | 120 | 1.000 | W=\mathfrak{P}^{C} | | |
| 0902.0431_FO2119 | 120 | 1.000 | P=(X, Y, \xi, \eta) | | |
| 0902.0431_FO2120 | 120 | 1.000 | W \ni P=(X, Y, \xi, \eta), X \neq 0 | | |
| 0902.0431_FO2121 | 120 | 0.991 | ((\mathrm{a})-(\mathrm{b})) \div 2,((\mathrm{a})-(\mathrm{c})) \div 8 | | |
| 0902.0431_FO2122 | 120 | 0.991 | (X, 0,-\xi, 2 \eta) \in W,(0,0, \xi, \eta) \in W | | |
| 0902.0431_FO2123 | 120 | 1.000 | (X, 0,0,3 \eta) \in W | | |
| 0902.0431_FO2124 | 120 | 1.000 | X_{1} \in \mathfrak{J}^{C} | | |
| 0902.0431_FO2125 | 120 | 1.000 | \left(X_{1}, X\right) \neq 0 | | |
| 0902.0431_FO2126 | 121 | 1.000 | P=(0, Y, \xi, \eta), Y \neq 0 | | |
| 0902.0431_FO2127 | 121 | 1.000 | B \times Y \neq 0 | | |
| 0902.0431_FO2128 | 121 | 1.000 | P=(0,0, \xi, \eta), \xi \neq 0 | | |
| 0902.0431_FO2129 | 121 | 1.000 | 0 \neq B \in \mathfrak{J}^{C} | | |
| 0902.0431_FO2130 | 121 | 0.979 | \mathfrak{e}_{7}{ }^{C} \mathfrak{P}^{C} | | |
| 0902.0431_FO2131 | 121 | 0.979 | \mathfrak{P}^{C}, \mathfrak{e}_{7}{ }^{C} \mathfrak{P}^{C}=\mathfrak{P}^{C} | | |
| 0902.0431_FO2132 | 121 | 0.859 | \Phi=\sum_{i}\left(P_{i} \times Q_{i}\right), \quad P_{i}, Q_{i} \in | | |
| 0902.0431_FO2133 | 121 | 1.000 | [\Phi, P \times Q]=\Phi P \times Q+P \times \Phi Q | | |
| 0902.0431_FO2134 | 121 | 1.000 | \mathfrak{a}=\left\{\sum_{i}\left(P_{i} \times\right.\right. | | |
| 0902.0431_FO2135 | 121 | 0.999 | \left.\left.Q_{i}\right) \mid P_{i}, Q_{i} \in \mathfrak{P}^{C}\right\} | | |
| 0902.0431_FO2136 | 121 | 0.971 | \left(\Phi_{1}, \Phi_{2}\right)_{7} | | |
| 0902.0431_FO2137 | 121 | 0.959 | \Phi_{i}=\Phi\left(\phi_{i}, A_{i}, B_{i}, \nu_{i}\right) \in \mathfrak{e}_{7}^{C} | | |
| 0902.0431_FO2138 | 121 | 0.761 | \Phi \in \mathfrak{e}_{7}{ }^{C}, P, Q \in \mathfrak{P}^{C} | | |
| 0902.0431_FO2139 | 121 | 1.000 | \left(\left[\Phi, \Phi_{1}\right], \Phi_{2}\right)_{7} | | |
| 0902.0431_FO2140 | 122 | 1.000 | B_{7} | | |
| 0902.0431_FO2141 | 122 | 0.682 | \Phi_{i}=\Phi\left(\phi_{i}, A_{i}, B_{i}, \nu_{i}\right) \in \mathfrak{e}_{7}{ }^{C} | | |
| 0902.0431_FO2142 | 122 | 1.000 | \Phi_{0}=\Phi_{1}=\Phi_{2}=\Phi(0,0,0,1) | | |
| 0902.0431_FO2143 | 123 | 1.000 | k=-9 | | |
| 0902.0431_FO2144 | 123 | 1.000 | P \in \mathfrak{P}^{C}, P \neq 0 | | |
| 0902.0431_FO2145 | 123 | 1.000 | Q \in \mathfrak{P}^{C} | | |
| 0902.0431_FO2146 | 123 | 1.000 | P \times Q \neq 0 | | |
| 0902.0431_FO2147 | 123 | 1.000 | P \times Q=0 | | |
| 0902.0431_FO2148 | 123 | 1.000 | 0=(\Phi, P \times Q)_{7}=(\Phi, Q \times P)_{7}=\{\Phi Q, P\} | | |
| 0902.0431_FO2149 | 123 | 1.000 | \mathfrak{e}_{7}{ }^{C} \mathfrak{P}^{C}=\mathfrak{P}^{C} | | |
| 0902.0431_FO2150 | 123 | 1.000 | \left\{\mathfrak{P}^{C}, P\right\}=0 | | |
| 0902.0431_FO2151 | 123 | 1.000 | P=0 | | |
| 0902.0431_FO2152 | 124 | 0.844 | \mathfrak{e}_{7}^{C} | | |
| 0902.0431_FO2153 | 124 | 1.000 | h_{\delta}=\sum_{k=0}^{3} \lambda_{k} H_{k}, H=\sum_{j=1}^{3} \mu_{j} E_{j} | | |
| 0902.0431_FO2154 | 124 | 1.000 | S \in \mathfrak{e}_{6}{ }^{C} \subset \mathfrak{e}_{7}{ }^{C} | | |
| 0902.0431_FO2155 | 124 | 0.999 | \pm\left(\mu_{j}+\frac{2}{3} \nu\right), 0 \leq j \leq 3 | | |
| 0902.0431_FO2156 | 125 | 1.000 | \pm \lambda_{k}-\frac{1}{2} \mu_{1}+\frac{2}{3} \nu, 0 \leq k \leq 3 | | |
| 0902.0431_FO2157 | 125 | 0.965 | \nu h_{\delta}, \kappa \pi h_{\delta} | | |
| 0902.0431_FO2158 | 125 | 1.000 | n_{1} n_{2} \cdots n_{7} | | |
| 0902.0431_FO2159 | 125 | 1.000 | \cdots+n_{7} \alpha_{7} | | |
| 0902.0431_FO2160 | 128 | 1.000 | \Pi=\left\{\alpha_{1}, \alpha_{2}, \cdots, \alpha_{7}\right\} | | |
| 0902.0431_FO2161 | 128 | 0.394 | h=\Phi\left(\sum_{k=0}^{3} \lambda_{k} H_{k}+\left(\sum_{j=1}^{3} \mu_{j} E_{j}\right)^{\sim}, 0,0, \nu\right), h^{\prime}=\Phi\left(\sum_{k=0}^{3} \lambda_{k}{ }^{\prime} H_{k}+\left(\sum_{j=1}^{3} \mu_{j}{ }^{\prime} E_{j}\right)^{\sim}, 0\right. | | |
| 0902.0431_FO2162 | 128 | 1.000 | \left.0, \nu^{\prime}\right) \in \mathfrak{h}_{\boldsymbol{R}} | | |
| 0902.0431_FO2163 | 128 | 1.000 | \alpha_{i}\left(B_{7}\left(H_{\alpha}, H\right)=\alpha(H), H \in\right. | | |
| 0902.0431_FO2164 | 128 | 0.508 | \mathfrak{h}) | | |
| 0902.0431_FO2165 | 128 | 1.000 | \alpha_{1}, \alpha_{2}, \cdots, \alpha_{7} | | |
| 0902.0431_FO2166 | 129 | 1.000 | T \oplus E_{6} | | |
| 0902.0431_FO2167 | 129 | 1.000 | A_{1} \oplus D_{6} | | |
| 0902.0431_FO2168 | 129 | 1.000 | A_{7} | | |
| 0902.0431_FO2169 | 129 | 1.000 | \lambda \gamma | | |
| 0902.0431_FO2170 | 129 | 1.000 | A_{2} \oplus A_{5} | | |
| 0902.0431_FO2171 | 129 | 1.000 | \left(E_{7}\right)_{(0,0,1,0)} | | |
| 0902.0431_FO2172 | 129 | 1.000 | \alpha(0,0,1,0)=(0,0,1,0) | | |
| 0902.0431_FO2173 | 129 | 1.000 | \alpha(0,0,0,1)=(0,0,0,1) | | |
| 0902.0431_FO2174 | 129 | 0.986 | \alpha \in E_{6}=\left\{\alpha \in \operatorname{Iso}_{C}\left(\mathfrak{J}^{C}\right) \mid \operatorname{det}(\alpha X)=\operatorname{det} X,\langle\alpha X\right. | | |
| 0902.0431_FO2175 | 129 | 0.996 | \alpha Y\rangle=\langle X, Y\rangle\} | | |
| 0902.0431_FO2176 | 130 | 1.000 | \widetilde{\alpha} \in E_{7} | | |
| 0902.0431_FO2177 | 130 | 0.999 | \langle\widetilde{\alpha} P, \widetilde{\alpha} Q\rangle=\langle P, Q\rangle | | |
| 0902.0431_FO2178 | 130 | 0.999 | \widetilde{\alpha} \in\left(E_{7}\right)_{(0,0,1,0)} | | |
| 0902.0431_FO2179 | 130 | 1.000 | \alpha(0,0,0,1)= | | |
| 0902.0431_FO2180 | 130 | 1.000 | 0 \neq \eta \in C | | |
| 0902.0431_FO2181 | 130 | 1.000 | \eta | | |
| 0902.0431_FO2182 | 130 | 1.000 | \alpha\left(\frac{1}{\xi}(Y \times Y), Y, \xi, \frac{1}{\xi^{2}} \operatorname{det} Y\right) \in \mathfrak{M}^{C} | | |
| 0902.0431_FO2183 | 130 | 1.000 | \epsilon=0 | | |
| 0902.0431_FO2184 | 131 | 1.000 | \beta \in E_{6}{ }^{C} | | |
| 0902.0431_FO2185 | 131 | 1.000 | \langle\alpha \dot{X}, \alpha \dot{Y}\rangle=\langle\dot{X}, \dot{Y}\rangle | | |
| 0902.0431_FO2186 | 131 | 1.000 | \langle\beta X, \beta Y\rangle= | | |
| 0902.0431_FO2187 | 131 | 1.000 | \beta_{1}=\tau \beta \tau | | |
| 0902.0431_FO2188 | 131 | 1.000 | X \times X | | |
| 0902.0431_FO2189 | 131 | 1.000 | \beta_{1} X=\tau \beta \tau X | | |
| 0902.0431_FO2190 | 131 | 1.000 | \tau \beta \tau | | |
| 0902.0431_FO2191 | 131 | 1.000 | \varphi_{1}(\theta): \mathfrak{P}^{C} \rightarrow \mathfrak{P}^{C} | | |
| 0902.0431_FO2192 | 131 | 1.000 | \varphi_{1}(\theta) \in E_{7}{ }^{C} | | |
| 0902.0431_FO2193 | 131 | 1.000 | U(1)=\{\theta \in | | |
| 0902.0431_FO2194 | 131 | 1.000 | C \mid(\tau \theta) \theta=1\} | | |
| 0902.0431_FO2195 | 131 | 1.000 | \varphi_{1}(\theta) \in E_{7} | | |
| 0902.0431_FO2196 | 131 | 1.000 | \left(E_{7}\right)_{0} | | |
| 0902.0431_FO2197 | 131 | 1.000 | a \in C | | |
| 0902.0431_FO2198 | 131 | 1.000 | \alpha_{i}(a): \mathfrak{P}^{C} \rightarrow \mathfrak{P}^{C}, i=1,2,3 | | |
| 0902.0431_FO2199 | 131 | 1.000 | \alpha_{i}(a) \in\left(E_{7}\right)_{0} | | |
| 0902.0431_FO2200 | 131 | 1.000 | p_{i}: \mathfrak{J}^{C} \rightarrow \mathfrak{J}^{C} | | |
| 0902.0431_FO2201 | 132 | 0.998 | \delta_{i j} | | |
| 0902.0431_FO2202 | 132 | 0.998 | \alpha_{1}\left(a_{1}\right), \alpha_{2}\left(a_{2}\right), \alpha\left(a_{3}\right),\left(a_{i} \in C\right) | | |
| 0902.0431_FO2203 | 132 | 1.000 | \alpha_{i}(a)=\exp \Phi_{i}(a) | | |
| 0902.0431_FO2204 | 132 | 1.000 | \left[\Phi_{i}\left(a_{i}\right), \Phi_{j}\left(a_{j}\right)\right]=0 | | |
| 0902.0431_FO2205 | 132 | 1.000 | \alpha_{i}\left(a_{i}\right) | | |
| 0902.0431_FO2206 | 132 | 1.000 | \alpha_{j}\left(a_{j}\right) | | |
| 0902.0431_FO2207 | 132 | 1.000 | P \in \mathfrak{M}^{C}, P \neq 0 | | |
| 0902.0431_FO2208 | 132 | 1.000 | \alpha \in\left(E_{7}\right)_{0} | | |
| 0902.0431_FO2209 | 132 | 1.000 | P=(X, Y, \xi, \eta) \in \mathfrak{M}^{C} | | |
| 0902.0431_FO2210 | 132 | 1.000 | \xi \neq 0 | | |
| 0902.0431_FO2211 | 132 | 1.000 | P=(X, Y, \xi, \eta), \xi \neq 0 | | |
| 0902.0431_FO2212 | 132 | 1.000 | X=\frac{1}{\xi}(Y \times Y) | | |
| 0902.0431_FO2213 | 132 | 1.000 | \tau \beta \tau Y | | |
| 0902.0431_FO2214 | 132 | 1.000 | P=(X, Y, 0, \eta), Y \neq 0 | | |
| 0902.0431_FO2215 | 132 | 0.887 | \tau \beta \tau Y \neq 0 | | |
| 0902.0431_FO2216 | 132 | 0.887 | \eta_{i} | | |
| 0902.0431_FO2217 | 132 | 0.887 | \eta_{i} \neq 0 | | |
| 0902.0431_FO2218 | 132 | 1.000 | \alpha_{i}(-\pi / 2) \in\left(E_{7}\right)_{0} | | |
| 0902.0431_FO2219 | 132 | 1.000 | \beta P | | |
| 0902.0431_FO2220 | 132 | 1.000 | P=(X, 0,0, \eta), X \neq 0 | | |
| 0902.0431_FO2221 | 132 | 1.000 | \beta X=\xi_{1} E_{1}+\xi_{2} E_{2}+\xi_{3} E_{3} | | |
| 0902.0431_FO2222 | 132 | 0.943 | \xi_{i} \in C | | |
| 0902.0431_FO2223 | 132 | 0.943 | \beta X \neq 0 | | |
| 0902.0431_FO2224 | 132 | 0.943 | \xi_{i} | | |
| 0902.0431_FO2225 | 132 | 0.943 | \xi_{i} \neq 0 | | |
| 0902.0431_FO2226 | 133 | 1.000 | P=(0,0,0, \eta), \eta \neq 0 | | |
| 0902.0431_FO2227 | 133 | 1.000 | \phi_{1}(\theta) \in U(1) \subset\left(E_{7}\right)_{0} | | |
| 0902.0431_FO2228 | 133 | 1.000 | \xi | | |
| 0902.0431_FO2229 | 133 | 1.000 | \xi>0 | | |
| 0902.0431_FO2230 | 133 | 1.000 | \mathfrak{M}_{1} | | |
| 0902.0431_FO2231 | 133 | 1.000 | P \in \mathfrak{M}_{1} | | |
| 0902.0431_FO2232 | 133 | 1.000 | \alpha P \in \mathfrak{M}_{1} | | |
| 0902.0431_FO2233 | 133 | 0.998 | (0,0,1,0) \in \mathfrak{M}_{1} | | |
| 0902.0431_FO2234 | 133 | 1.000 | \langle\alpha P, \alpha P\rangle=\langle P, P\rangle=1 | | |
| 0902.0431_FO2235 | 133 | 1.000 | r_{1}, r_{2}, r_{3} \in \boldsymbol{R}, 0 \leq r_{i}<\frac{\pi}{2} | | |
| 0902.0431_FO2236 | 134 | 0.952 | \eta_{i}=0 | | |
| 0902.0431_FO2237 | 134 | 1.000 | \alpha P | | |
| 0902.0431_FO2238 | 134 | 1.000 | =\alpha_{3}\left(a_{3}\right) \alpha_{2}\left(a_{2}\right) \alpha_{1}\left(a_{1}\right)(0,0,1,0) | | |
| 0902.0431_FO2239 | 134 | 1.000 | \mathfrak{M}_{1}=\left(E_{7}\right)_{0}(0,0,1,0), \mathfrak{M}_{1} | | |
| 0902.0431_FO2240 | 134 | 1.000 | E_{7} / E_{6} \simeq \mathfrak{M}_{1} | | |
| 0902.0431_FO2241 | 134 | 1.000 | \alpha \in z\left(E_{7}\right) | | |
| 0902.0431_FO2242 | 134 | 1.000 | \beta \in E_{6} \subset E_{7} | | |
| 0902.0431_FO2243 | 134 | 1.000 | \beta \alpha(0,0,1,0)=\alpha \beta(0,0,1,0)=\alpha(0,0,1,0) | | |
| 0902.0431_FO2244 | 134 | 1.000 | \alpha(0,0,1,0)=(X, Y, \xi, \eta) \in | | |
| 0902.0431_FO2245 | 134 | 1.000 | (\beta X, \tau \beta \tau Y, \xi, \eta)=(X, Y, \xi, \eta) | | |
| 0902.0431_FO2246 | 135 | 1.000 | X=Y=0 | | |
| 0902.0431_FO2247 | 135 | 1.000 | \alpha(0,0,1,0) | | |
| 0902.0431_FO2248 | 135 | 1.000 | \alpha(0,0,1,0) \in \mathfrak{M}^{C} | | |
| 0902.0431_FO2249 | 135 | 1.000 | \xi \eta=0 | | |
| 0902.0431_FO2250 | 135 | 1.000 | \xi=0 | | |
| 0902.0431_FO2251 | 135 | 1.000 | \alpha(0,0,1,0)=(0,0,0, \eta), \eta \neq 0 | | |
| 0902.0431_FO2252 | 135 | 1.000 | \varphi_{1}(\theta) \in U(1) \subset E_{7} | | |
| 0902.0431_FO2253 | 135 | 1.000 | \theta^{-3} \eta=\theta^{3} \eta | | |
| 0902.0431_FO2254 | 135 | 1.000 | \theta | | |
| 0902.0431_FO2255 | 135 | 1.000 | \xi \neq 0, \eta=0 | | |
| 0902.0431_FO2256 | 135 | 1.000 | \alpha(0,0,1,0)=(0,0, \xi, 0) | | |
| 0902.0431_FO2257 | 135 | 1.000 | \alpha(0,0,0,1)=(0,0,0, \zeta) | | |
| 0902.0431_FO2258 | 135 | 1.000 | \{\alpha(0,0,1,0), \alpha(0,0,1,0)\}=1 | | |
| 0902.0431_FO2259 | 135 | 1.000 | \xi \zeta=1 | | |
| 0902.0431_FO2260 | 135 | 1.000 | \xi=\xi^{-1} | | |
| 0902.0431_FO2261 | 135 | 1.000 | \xi= \pm 1 | | |
| 0902.0431_FO2262 | 135 | 1.000 | \xi=1 | | |
| 0902.0431_FO2263 | 135 | 1.000 | \alpha \in z\left(E_{6}\right)=\left\{1, \omega 1, \omega^{2} 1\right\} | | |
| 0902.0431_FO2264 | 135 | 1.000 | \omega^{\prime}=\omega^{\prime-1} | | |
| 0902.0431_FO2265 | 135 | 1.000 | \omega^{\prime}=1 | | |
| 0902.0431_FO2266 | 135 | 1.000 | \xi=-1 | | |
| 0902.0431_FO2267 | 135 | 1.000 | -\alpha \in z\left(E_{6}\right) | | |
| 0902.0431_FO2268 | 135 | 1.000 | -\alpha=1 | | |
| 0902.0431_FO2269 | 135 | 1.000 | z\left(E_{7}\right)=\{1,-1\} | | |
| 0902.0431_FO2270 | 135 | 1.000 | E_{7}=\left\{\alpha \in \operatorname{Iso}_{C}\left(\mathfrak{P}^{C}\right) \mid \alpha(P \times Q) \alpha^{-1}=\alpha P \times \alpha Q,\langle\alpha P, \alpha Q\rangle=\right. | | |
| 0902.0431_FO2271 | 135 | 0.983 | \langle P, Q\rangle\} | | |
| 0902.0431_FO2272 | 135 | 0.857 | \operatorname{subgroup}\left(U(1) \times E_{6}\right) / \boldsymbol{Z}_{3} | | |
| 0902.0431_FO2273 | 136 | 1.000 | \iota=\varphi_{1}(i) \in U(1) \subset E_{7}, \iota^{2}=-1 \in z\left(E_{7}\right) | | |
| 0902.0431_FO2274 | 136 | 1.000 | \iota^{4}=1 | | |
| 0902.0431_FO2275 | 136 | 1.000 | \delta=\alpha_{1}\left(\frac{i \pi}{4}\right) \alpha_{2}\left(\frac{i \pi}{4}\right) \alpha_{3}\left(\frac{i \pi}{4}\right) | | |
| 0902.0431_FO2276 | 136 | 1.000 | \iota=\delta^{-1} \lambda \delta | | |
| 0902.0431_FO2277 | 136 | 0.562 | \widetilde{\iota}: E_{7} \rightarrow E_{7} | | |
| 0902.0431_FO2278 | 136 | 1.000 | \left(E_{7}\right)^{\iota} | | |
| 0902.0431_FO2279 | 136 | 1.000 | \left(E_{7}\right)^{\iota} \cong\left(U(1) \times E_{6}\right) / \boldsymbol{Z}_{3}, \quad \boldsymbol{Z}_{3}=\left\{(1,1),(\omega, \omega 1),\left(\omega^{2}, \omega^{2} 1\right)\right\} | | |
| 0902.0431_FO2280 | 136 | 0.999 | \omega=-\frac{1}{2}+\frac{\sqrt{3}}{2} i \in C | | |
| 0902.0431_FO2281 | 136 | 1.000 | \varphi: U(1) \times E_{6} \rightarrow\left(E_{7}\right)^{\iota} | | |
| 0902.0431_FO2282 | 136 | 1.000 | \varphi(\theta, \beta) \in\left(E_{7}\right)^{\iota} | | |
| 0902.0431_FO2283 | 136 | 1.000 | \varphi_{1}(\theta) | | |
| 0902.0431_FO2284 | 136 | 1.000 | \alpha \in\left(E_{7}\right)^{\iota} | | |
| 0902.0431_FO2285 | 136 | 1.000 | \iota \alpha=\alpha \iota, \alpha | | |
| 0902.0431_FO2286 | 136 | 1.000 | \alpha(0,0,1,0), \alpha(0,0,0,1) \in \mathfrak{M}^{C} | | |
| 0902.0431_FO2287 | 136 | 1.000 | M=N=0 | | |
| 0902.0431_FO2288 | 136 | 1.000 | M \neq 0, \mu=0 | | |
| 0902.0431_FO2289 | 136 | 1.000 | \{\alpha(0,0,1,0), \alpha(0,0,0,1)\}=\{(0,0,1,0),(0,0,0,1)\}=1 | | |
| 0902.0431_FO2290 | 136 | 1.000 | N \neq 0, \nu=0 | | |
| 0902.0431_FO2291 | 137 | 1.000 | a(X) \operatorname{det} X=a(X)^{2} b(X \times X) | | |
| 0902.0431_FO2292 | 137 | 1.000 | \mu=0 | | |
| 0902.0431_FO2293 | 137 | 1.000 | a \neq 0 | | |
| 0902.0431_FO2294 | 137 | 1.000 | a: \mathfrak{J}^{C} \rightarrow C | | |
| 0902.0431_FO2295 | 137 | 0.993 | \left\{X \in \mathfrak{J}^{C} \mid a(X) \neq 0\right\} | | |
| 0902.0431_FO2296 | 137 | 0.993 | b(X \times X) | | |
| 0902.0431_FO2297 | 137 | 1.000 | a(X)=0 | | |
| 0902.0431_FO2298 | 137 | 1.000 | M=0 | | |
| 0902.0431_FO2299 | 137 | 1.000 | N=0 | | |
| 0902.0431_FO2300 | 137 | 1.000 | \alpha(0,0,1,0)=(0,0, \mu, 0) | | |
| 0902.0431_FO2301 | 137 | 0.627 | \alpha(0,0,0,1)=(0,0,0, \nu) | | |
| 0902.0431_FO2302 | 137 | 0.627 | \{\alpha \mathrm{i}, \alpha \underline{1}\}=1,\langle\alpha \mathrm{i}, \alpha \mathrm{i}\rangle=1 | | |
| 0902.0431_FO2303 | 137 | 1.000 | \theta^{3}=\mu | | |
| 0902.0431_FO2304 | 137 | 1.000 | \beta=\varphi_{1}(\theta)^{-1} \alpha | | |
| 0902.0431_FO2305 | 137 | 1.000 | \beta(0,0,1,0)= | | |
| 0902.0431_FO2306 | 137 | 1.000 | (0,0,1,0), \beta(0,0,0,1)=(0,0,0,1) | | |
| 0902.0431_FO2307 | 138 | 0.966 | \operatorname{Ker} \varphi=\left\{(1,1),(\omega, \omega 1),\left(\omega^{2}, \omega^{2} 1\right)\right\}=\boldsymbol{Z}_{3} | | |
| 0902.0431_FO2308 | 138 | 0.999 | \left(U(1) \times E_{6}\right) / \boldsymbol{Z}_{3} \cong\left(E_{7}\right)^{\iota} | | |
| 0902.0431_FO2309 | 138 | 1.000 | \varphi: U(1) \times E_{6} \rightarrow\left(E_{6}\right)^{\iota} | | |
| 0902.0431_FO2310 | 138 | 1.000 | \varphi_{*}: \mathfrak{u}(1) \oplus \mathfrak{e}_{6} \rightarrow\left(\mathfrak{e}_{7}\right)^{\iota} | | |
| 0902.0431_FO2312 | 138 | 1.000 | \sigma \in F_{4} \subset E_{6} \subset E_{7} | | |
| 0902.0431_FO2313 | 138 | 1.000 | \left(E_{7}\right)^{\sigma} | | |
| 0902.0431_FO2314 | 138 | 1.000 | \kappa, \mu: \mathfrak{P}^{C} \rightarrow \mathfrak{P}^{C} | | |
| 0902.0431_FO2315 | 138 | 1.000 | \mu | | |
| 0902.0431_FO2316 | 138 | 0.997 | \quad \kappa \mu=\mu \kappa, \quad\left\{\begin{array}{l}\kappa \sigma=\sigma \kappa \\ \mu \sigma=\sigma \mu,\end{array} \quad\left\{\begin{array}{l}\kappa \lambda=-\lambda \kappa \\ \mu \lambda=-\lambda \mu .\end{array}\right.\right. | | |
| 0902.0431_FO2317 | 139 | 1.000 | \kappa \alpha=\alpha \kappa | | |
| 0902.0431_FO2318 | 139 | 1.000 | \sigma=\exp \pi i \kappa | | |
| 0902.0431_FO2319 | 139 | 1.000 | \sigma \alpha=(\exp \pi i \kappa) \alpha=\alpha(\exp \pi i \kappa)=\alpha \sigma | | |
| 0902.0431_FO2320 | 139 | 1.000 | \left(E_{7}\right)^{\kappa, \mu} | | |
| 0902.0431_FO2321 | 139 | 1.000 | \left(\left(E_{7}\right)^{\kappa, \mu}\right)_{\left(0, E_{1}, 0,1\right)} | | |
| 0902.0431_FO2322 | 139 | 1.000 | \left(\mathfrak{e}_{7}\right)^{\sigma},\left(\mathfrak{e}_{7}\right)^{\kappa, \mu},\left(\left(\mathfrak{e}_{7}\right)^{\kappa, \mu}\right)_{\left(0, E_{1}, 0,1\right)} | | |
| 0902.0431_FO2323 | 139 | 1.000 | \left(E_{7}\right)^{\sigma},\left(E_{7}\right)^{\kappa, \mu},\left(\left(E_{7}\right)^{\kappa, \mu}\right)_{\left(0, E_{1}, 0,1\right)} | | |
| 0902.0431_FO2324 | 139 | 1.000 | \Phi=\Phi(\phi, A,-\tau A, \nu) \in \mathfrak{e}_{7} | | |
| 0902.0431_FO2325 | 139 | 1.000 | \kappa \Phi=\Phi \kappa | | |
| 0902.0431_FO2326 | 139 | 1.000 | \mu \Phi=\Phi \mu | | |
| 0902.0431_FO2327 | 139 | 1.000 | \kappa \Phi P=\Phi \kappa P, P=(X, Y, \xi, \eta) \in \mathfrak{P}^{C} | | |
| 0902.0431_FO2328 | 139 | 0.915 | Y=E_{1} | | |
| 0902.0431_FO2329 | 139 | 0.915 | \left(A, E_{1}\right)=0 | | |
| 0902.0431_FO2330 | 139 | 1.000 | \phi \in\left(\mathfrak{e}_{6}\right)^{\sigma} | | |
| 0902.0431_FO2331 | 139 | 1.000 | \phi E_{1}=k E_{1}, k \in i \boldsymbol{R} | | |
| 0902.0431_FO2332 | 139 | 1.000 | X=E_{1} | | |
| 0902.0431_FO2333 | 139 | 1.000 | k=-\frac{2}{3} \nu | | |
| 0902.0431_FO2334 | 139 | 0.976 | A \in\left(\mathfrak{J}^{C}\right)_{\sigma} | | |
| 0902.0431_FO2335 | 139 | 0.976 | \kappa_{1}(A \times X)=\kappa_{1} A \times \kappa_{1} X, X \in \mathfrak{J}^{C} | | |
| 0902.0431_FO2336 | 139 | 1.000 | \kappa_{1} \phi=\phi \kappa_{1} | | |
| 0902.0431_FO2337 | 139 | 1.000 | A \in\left(\mathfrak{J}^{C}\right)_{\sigma},\left(E_{1}, A\right)=0 | | |
| 0902.0431_FO2338 | 139 | 1.000 | \kappa_{1} A=-A | | |
| 0902.0431_FO2339 | 140 | 1.000 | \Phi=\Phi(\phi, A,-\tau A, \nu) \in\left(\mathfrak{e}_{7}\right)^{\kappa, \mu} | | |
| 0902.0431_FO2340 | 140 | 1.000 | \Phi\left(\left(0, E_{1}, 0,1\right)\right)=0 | | |
| 0902.0431_FO2341 | 140 | 1.000 | \nu=\tau \nu | | |
| 0902.0431_FO2342 | 140 | 1.000 | \tau \nu=-\nu | | |
| 0902.0431_FO2343 | 140 | 1.000 | \nu=0 | | |
| 0902.0431_FO2344 | 140 | 1.000 | \phi E_{1}=0 | | |
| 0902.0431_FO2345 | 140 | 1.000 | 2 A \times E_{1}=\tau A | | |
| 0902.0431_FO2346 | 140 | 1.000 | \left(E_{1}, A\right)=0 | | |
| 0902.0431_FO2347 | 140 | 1.000 | \nu \in i \boldsymbol{R} | | |
| 0902.0431_FO2348 | 140 | 1.000 | \phi(\nu): \mathfrak{J}^{C} \rightarrow \mathfrak{J}^{C} | | |
| 0902.0431_FO2349 | 140 | 1.000 | \phi(\nu) \in\left(\mathfrak{e}_{6}\right)^{\sigma} | | |
| 0902.0431_FO2350 | 140 | 0.999 | \mathfrak{a}_{1}=\left\{\Phi\left(\phi(\nu), a E_{1},-\tau a E_{1}, \nu\right) \mid a \in C, \nu \in i \boldsymbol{R}\right\} | | |
| 0902.0431_FO2351 | 140 | 1.000 | \left(\mathfrak{e}_{7}\right)^{\sigma} | | |
| 0902.0431_FO2352 | 140 | 1.000 | \mathfrak{s u}(2) | | |
| 0902.0431_FO2353 | 140 | 1.000 | \mathfrak{a}_{1} | | |
| 0902.0431_FO2354 | 140 | 1.000 | \left(\mathfrak{e}_{7}\right)^{\kappa, \mu} | | |
| 0902.0431_FO2355 | 140 | 1.000 | \varphi_{*}: \mathfrak{a}_{1} \rightarrow \mathfrak{s u}(2)=\left\{D \in M(2, C) \mid \tau\left({ }^{t} D\right)=-D\right\} | | |
| 0902.0431_FO2356 | 140 | 0.993 | \nu^{\prime}=\frac{1}{3} \nu+\frac{1}{2}\left(E_{1}, \phi E_{1}\right), a=\left(E_{1}, A\right) | | |
| 0902.0431_FO2357 | 141 | 1.000 | \alpha_{i}(a) \in E_{7}, i=2,3 | | |
| 0902.0431_FO2358 | 141 | 1.000 | \Phi\left(0,-\tau a E_{i}, a E_{i}, 0\right) \in\left(\mathfrak{e}_{7}\right)^{\kappa, \mu} | | |
| 0902.0431_FO2359 | 141 | 1.000 | \alpha_{i}(a)=\exp \Phi\left(0,-\tau a E_{i}\right. | | |
| 0902.0431_FO2360 | 141 | 0.999 | \left.a E_{i}, 0\right) \in\left(E_{7}\right)^{\kappa, \mu}, i=2,3 | | |
| 0902.0431_FO2361 | 141 | 0.999 | \alpha_{2}(a) | | |
| 0902.0431_FO2362 | 141 | 0.999 | \alpha_{3}(\tau a) | | |
| 0902.0431_FO2363 | 141 | 1.000 | \Phi\left(0,-\tau a E_{2}-a E_{3}, a E_{2}+\tau a E_{3}, 0\right) \in\left(\left(\mathfrak{e}_{7}\right)^{\kappa, \mu}\right)_{\left(0, E_{1}, 0,1\right)} | | |
| 0902.0431_FO2364 | 141 | 1.000 | \alpha_{23}(a) \in | | |
| 0902.0431_FO2365 | 141 | 1.000 | \left(\left(E_{7}\right)^{\kappa, \nu}\right)_{\left(0, E_{1}, 0,1\right)} | | |
| 0902.0431_FO2366 | 141 | 1.000 | \alpha \in\left(\left(E_{7}\right)^{\kappa, \mu}\right)_{\left(0, E_{1}, 0,1\right)} | | |
| 0902.0431_FO2367 | 141 | 1.000 | \alpha \in\left(E_{7}\right)^{\kappa, \mu} | | |
| 0902.0431_FO2368 | 141 | 1.000 | \alpha\left(0, E_{1}, 0,1\right)=\left(0, E_{1}, 0,1\right) | | |
| 0902.0431_FO2369 | 141 | 1.000 | \alpha\left(0,-E_{1}, 0,1\right)= | | |
| 0902.0431_FO2370 | 141 | 0.989 | \left(0,-E_{1}, 0,1\right) | | |
| 0902.0431_FO2371 | 141 | 0.989 | \alpha\left(0, E_{1}, 0,0\right)=\left(0, E_{1}, 0,0\right) | | |
| 0902.0431_FO2372 | 141 | 1.000 | \alpha(0,0,1,0)=\alpha \mu\left(0, E_{1}, 0,0\right)=\mu \alpha\left(0, E_{1}, 0,0\right)=\mu\left(0, E_{1}, 0,0\right)= | | |
| 0902.0431_FO2373 | 141 | 0.957 | \alpha(0,0,1,0)= | | |
| 0902.0431_FO2374 | 141 | 0.966 | (0,0,1,0) | | |
| 0902.0431_FO2375 | 141 | 1.000 | \alpha \in \operatorname{Spin}(10) | | |
| 0902.0431_FO2376 | 141 | 1.000 | V^{11} | | |
| 0902.0431_FO2377 | 142 | 1.000 | (P, P)_{\mu} | | |
| 0902.0431_FO2378 | 142 | 0.985 | \quad\left(\left(E_{7}\right)^{\kappa, \mu}\right)_{\left(0, E_{1}, 0,1\right)} / \operatorname{Spin}(10) \simeq S^{10} | | |
| 0902.0431_FO2379 | 142 | 1.000 | S^{10}=\left\{P \in V^{11} \mid(P, P)_{\mu}=1\right\} | | |
| 0902.0431_FO2380 | 142 | 1.000 | \alpha \in | | |
| 0902.0431_FO2381 | 142 | 1.000 | P \in S^{10} | | |
| 0902.0431_FO2382 | 142 | 1.000 | \alpha P \in S^{10} | | |
| 0902.0431_FO2383 | 142 | 1.000 | S^{10} | | |
| 0902.0431_FO2384 | 142 | 1.000 | \left(0,-i E_{1}, 0, i\right) \in S^{10} | | |
| 0902.0431_FO2385 | 142 | 1.000 | a \in \boldsymbol{R}, 0 \leq a<\frac{\pi}{4} | | |
| 0902.0431_FO2386 | 142 | 1.000 | \tau \xi-\xi=0 | | |
| 0902.0431_FO2387 | 142 | 1.000 | a=\frac{\pi}{4} | | |
| 0902.0431_FO2388 | 142 | 1.000 | \alpha_{23}(a) | | |
| 0902.0431_FO2389 | 142 | 1.000 | \alpha_{23}(a) P | | |
| 0902.0431_FO2390 | 142 | 1.000 | \beta \in \operatorname{Spin}(10)=\left(E_{6}\right)_{E_{1}} \subset\left(\left(E_{7}\right)^{\kappa, \mu}\right)_{\left(0, E_{1}, 0,1\right)} | | |
| 0902.0431_FO2391 | 142 | 1.000 | \alpha_{23}(-\pi / 4) \in\left(\left(E_{7}\right)^{\kappa, \mu}\right)_{\left(0, E_{1}, 0,1\right)} | | |
| 0902.0431_FO2392 | 142 | 0.999 | \left(0-i E_{1}, 0, i\right) | | |
| 0902.0431_FO2393 | 142 | 0.999 | \left(\left(E_{7}\right)^{\kappa, \mu}\right)_{\left(0, E_{1}, 0,1\right)} / \operatorname{Spin}(10) \simeq S^{10} | | |
| 0902.0431_FO2394 | 142 | 0.506 | \quad\left(\left(E_{7}\right)^{\kappa, \mu}\right)_{\left(0, E_{1}, 0,1\right)} \cong \operatorname{Spin}(11) | | |
| 0902.0431_FO2395 | 143 | 1.000 | \left(\left(E_{7}\right)^{\kappa, \mu}\right)_{\left(0, E_{1}, 0,1\right)} \rightarrow S O(11)=S O\left(V^{11}\right) | | |
| 0902.0431_FO2396 | 143 | 1.000 | p(\alpha)=\alpha \mid V^{11} | | |
| 0902.0431_FO2397 | 143 | 0.995 | p^{\prime}: \operatorname{Spin}(10) \rightarrow S O(10)=S O\left(V^{10}\right) | | |
| 0902.0431_FO2398 | 143 | 1.000 | V^{10}=\left\{P \in V^{11} \mid P=(X, 0,0,0)\right\} | | |
| 0902.0431_FO2399 | 143 | 1.000 | p^{\prime}: \operatorname{Spin}(10) \rightarrow S O(10) | | |
| 0902.0431_FO2400 | 143 | 1.000 | p:\left(\left(E_{7}\right)^{\kappa, \mu}\right)_{\left(0, E_{1}, 0,1\right)} \rightarrow S O(11) | | |
| 0902.0431_FO2401 | 143 | 0.989 | \operatorname{Ker} p | | |
| 0902.0431_FO2402 | 143 | 0.989 | \operatorname{Ker} p^{\prime} | | |
| 0902.0431_FO2403 | 143 | 1.000 | \operatorname{Spin}(11) | | |
| 0902.0431_FO2404 | 143 | 1.000 | S O(11) | | |
| 0902.0431_FO2405 | 143 | 1.000 | \alpha(t): \mathfrak{P}^{C} \rightarrow \mathfrak{P}^{C} | | |
| 0902.0431_FO2406 | 143 | 1.000 | \alpha(t) \in\left(E_{7}\right)^{\kappa, \mu} | | |
| 0902.0431_FO2407 | 143 | 1.000 | \nu=i t \in i \boldsymbol{R} | | |
| 0902.0431_FO2408 | 143 | 1.000 | \phi(\nu)=2 \nu E_{1} \vee E_{1} \in\left(\mathfrak{e}_{6}\right)^{\sigma} | | |
| 0902.0431_FO2409 | 143 | 1.000 | \Phi(\phi(\nu), 0,0,-2 \nu) \in | | |
| 0902.0431_FO2410 | 143 | 0.999 | \left(\mathfrak{e}_{7}\right)^{\kappa, \nu} | | |
| 0902.0431_FO2411 | 143 | 0.999 | \alpha(t)=\exp \Phi(\phi(\nu), 0,0,-2 \nu) | | |
| 0902.0431_FO2412 | 143 | 0.999 | \alpha(t) \in | | |
| 0902.0431_FO2413 | 143 | 1.000 | V^{12} | | |
| 0902.0431_FO2414 | 143 | 1.000 | \left(E_{7}\right)^{\kappa, \mu} / \operatorname{Spin}(11) \simeq S^{11} | | |
| 0902.0431_FO2415 | 144 | 1.000 | S^{11}=\left\{P \in V^{12} \mid(P, P)_{\mu}=1\right\} | | |
| 0902.0431_FO2416 | 144 | 1.000 | P \in S^{11} | | |
| 0902.0431_FO2417 | 144 | 1.000 | \alpha P \in S^{11} | | |
| 0902.0431_FO2418 | 144 | 1.000 | S^{11} | | |
| 0902.0431_FO2419 | 144 | 1.000 | \left(0, E_{1}, 0,1\right) \in S^{11} | | |
| 0902.0431_FO2420 | 144 | 1.000 | e^{-2 i t} \eta \in i \boldsymbol{R} | | |
| 0902.0431_FO2421 | 144 | 1.000 | \alpha(t) | | |
| 0902.0431_FO2422 | 144 | 1.000 | \beta \in \operatorname{Spin}(11)=\left(\left(E_{7}\right)^{\kappa, \mu}\right)_{\left(0, E_{1}, 0,1\right)} | | |
| 0902.0431_FO2423 | 144 | 1.000 | \alpha(-\pi / 4) \in\left(E_{7}\right)^{\kappa, \mu} | | |
| 0902.0431_FO2424 | 144 | 1.000 | \left(0, E_{1}, 0,1\right) | | |
| 0902.0431_FO2425 | 144 | 0.993 | \left(E_{7}\right)^{\kappa, \mu} / \operatorname{Spin}(11) | | |
| 0902.0431_FO2426 | 144 | 1.000 | \simeq S^{11} | | |
| 0902.0431_FO2427 | 144 | 1.000 | \quad\left(E_{7}\right)^{\kappa, \mu} \cong \operatorname{Spin}(12) | | |
| 0902.0431_FO2428 | 144 | 1.000 | p(\alpha)=\alpha \mid V^{12} | | |
| 0902.0431_FO2429 | 144 | 1.000 | \left(E_{6}\right)^{\kappa, \mu} | | |
| 0902.0431_FO2430 | 144 | 0.920 | p^{\prime}: \operatorname{Spin}(11) \rightarrow S O(11) | | |
| 0902.0431_FO2431 | 144 | 1.000 | p:\left(E_{7}\right)^{\kappa, \mu} \rightarrow S O(12) | | |
| 0902.0431_FO2432 | 145 | 0.988 | \operatorname{Spin}(12) | | |
| 0902.0431_FO2433 | 145 | 1.000 | S O(12) | | |
| 0902.0431_FO2434 | 145 | 0.484 | z( | | |
| 0902.0431_FO2435 | 145 | 0.979 | S U(2)=\left\{\left.A \in M(2, C)\right|^{t}(\tau A) A=\right. | | |
| 0902.0431_FO2436 | 145 | 0.988 | E, \operatorname{det} A=1\} | | |
| 0902.0431_FO2437 | 145 | 0.988 | A \in S U(2) | | |
| 0902.0431_FO2438 | 145 | 0.988 | \varphi_{2}(A): \mathfrak{P}^{C} \rightarrow \mathfrak{P}^{C} | | |
| 0902.0431_FO2439 | 145 | 0.998 | \Phi\left(\phi(\nu), a E_{1},-\tau a E_{1}, \nu\right) \in \mathfrak{a}_{1}\left(\phi(\nu)=2 \nu E_{1} \vee E_{1}, \nu \in i \boldsymbol{R}, a \in\right. | | |
| 0902.0431_FO2440 | 145 | 0.998 | A=\exp \left(\begin{array}{cc}\nu & a \\ -\tau a & -\nu\end{array}\right) \in S U(2) | | |
| 0902.0431_FO2441 | 146 | 0.707 | \left(E_{7}\right)^{\sigma} \cong(S U(2) \times \operatorname{Spin}(12)) / \boldsymbol{Z}_{2}, \boldsymbol{Z}_{2}=\{(E, 1),(-E,-\sigma)\} | | |
| 0902.0431_FO2442 | 146 | 0.947 | \varphi: \operatorname{SU}(2) \times \operatorname{Spin}(12) \rightarrow\left(E_{7}\right)^{\sigma} | | |
| 0902.0431_FO2443 | 146 | 0.768 | S U(2) | | |
| 0902.0431_FO2444 | 146 | 0.840 | \varphi_{2}(A) \in S U(2) | | |
| 0902.0431_FO2445 | 146 | 0.840 | \beta \in \operatorname{Spin}(12) | | |
| 0902.0431_FO2446 | 146 | 1.000 | \varphi_{2}(A) \beta=\beta \varphi_{2}(A) | | |
| 0902.0431_FO2447 | 146 | 1.000 | \varphi_{*}: \mathfrak{a}_{1} \oplus\left(\mathfrak{e}_{7}\right)^{\kappa, \mu} \rightarrow\left(\mathfrak{e}_{7}\right)^{\sigma} | | |
| 0902.0431_FO2448 | 146 | 1.000 | \operatorname{Ker} \varphi=\left\{(E, 1),\left(-E, \varphi_{2}(-E)\right)\right\}= | | |
| 0902.0431_FO2449 | 146 | 1.000 | \{(E, 1),(-E,-\sigma)\}=\boldsymbol{Z}_{2} | | |
| 0902.0431_FO2450 | 146 | 0.792 | (S U(2) \times \operatorname{Spin}(12)) / \boldsymbol{Z}_{2} \cong\left(E_{7}\right)^{\sigma} | | |
| 0902.0431_FO2451 | 146 | 1.000 | X \in\left(\mathfrak{J}^{C}\right)_{\sigma} | | |
| 0902.0431_FO2452 | 146 | 1.000 | \xi_{2} \geq 0, \xi_{3} \geq 0 | | |
| 0902.0431_FO2453 | 146 | 1.000 | i\left(E_{1}-E_{2}\right)^{\sim}, i\left(E_{1}-E_{3}\right)^{\sim}, i \widetilde{F}_{1}(a), \widetilde{A}_{1}(a)(a \in \mathfrak{C}) \in\left(\mathfrak{e}_{6}\right)^{\sigma} | | |
| 0902.0431_FO2454 | 146 | 1.000 | P \in\left(\mathfrak{M}^{C}\right)_{\sigma}=\left\{P \in \mathfrak{M}^{C} \mid \sigma P=P\right\} | | |
| 0902.0431_FO2455 | 146 | 1.000 | \alpha \in\left(\left(E_{7}\right)^{\sigma}\right)_{0} | | |
| 0902.0431_FO2456 | 146 | 1.000 | \Phi\left(0,-\tau a E_{i}, a E_{i}, 0\right) \in\left(\mathfrak{e}_{7}\right)^{\sigma}, i=1,2,3 | | |
| 0902.0431_FO2457 | 146 | 0.926 | \quad\left(E_{7}\right)^{\sigma} /\left(E_{6}\right)^{\sigma} \simeq\left(\mathfrak{M}_{1}\right)_{\sigma}=\left\{P \in \mathfrak{M}_{1} \mid \sigma P=P\right\} | | |
| 0902.0431_FO2458 | 146 | 1.000 | \alpha_{i}(a) | | |
| 0902.0431_FO2459 | 146 | 1.000 | \left(\left(E_{7}\right)^{\sigma}\right)_{0} | | |
| 0902.0431_FO2460 | 146 | 0.909 | \left(E_{6}\right)^{\sigma} \cong(U(1) \times \operatorname{Spin}(10)) / \boldsymbol{Z}_{4}( | | |
| 0902.0431_FO2461 | 146 | 0.909 | \left(\mathfrak{M}_{1}\right)_{\sigma} | | |
| 0902.0431_FO2462 | 147 | 0.998 | \gamma \in G_{2} \subset F_{4} \subset E_{6} \subset E_{7} | | |
| 0902.0431_FO2463 | 147 | 1.000 | \left(E_{7}\right)^{\tau \gamma} | | |
| 0902.0431_FO2464 | 147 | 1.000 | \left(\mathfrak{P}^{C}\right)_{\tau \gamma},\left(\mathfrak{P}^{C}\right)_{-\tau \gamma} | | |
| 0902.0431_FO2465 | 147 | 0.999 | \left(\mathfrak{P}^{C}\right)_{\tau \gamma}: \mathfrak{P}^{C}=\left(\left(\mathfrak{P}^{C}\right)_{\tau \gamma}\right)^{C} | | |
| 0902.0431_FO2466 | 147 | 1.000 | k: M(4, \boldsymbol{H}) \rightarrow | | |
| 0902.0431_FO2467 | 147 | 0.989 | M(8, \boldsymbol{C}) | | |
| 0902.0431_FO2468 | 147 | 1.000 | B \in \mathfrak{s u}(8) | | |
| 0902.0431_FO2469 | 147 | 0.931 | D_{1}=\frac{B-J \bar{B} J}{2}, T_{1}=\frac{B+J \bar{B} J}{2 e_{1}} \in M(8, C) | | |
| 0902.0431_FO2470 | 147 | 1.000 | D=k^{-1}\left(D_{1}\right), T=k^{-1}\left(T_{1}\right) \in M(4, \boldsymbol{H}) | | |
| 0902.0431_FO2471 | 148 | 1.000 | J D_{1}+e_{1} J T_{1}=0 | | |
| 0902.0431_FO2472 | 148 | 1.000 | \bar{D}_{1} J+e_{1} \bar{T}_{1} J=0 | | |
| 0902.0431_FO2473 | 148 | 1.000 | D_{1} J-e_{1} T_{1} J=0 | | |
| 0902.0431_FO2474 | 148 | 1.000 | D_{1}-e_{1} T_{1}=0 | | |
| 0902.0431_FO2475 | 148 | 0.998 | D_{1}=T_{1}=0 | | |
| 0902.0431_FO2476 | 148 | 1.000 | g: \mathfrak{J}^{C} \rightarrow \mathfrak{J}(4, \boldsymbol{H})^{C}, g(M+\boldsymbol{a})= | | |
| 0902.0431_FO2477 | 148 | 0.998 | \left(\begin{array}{cc}\frac{1}{2} \operatorname{tr}(M) & i \boldsymbol{a} \\ i \boldsymbol{a}^{*} & M-\frac{1}{2} \operatorname{tr}(M) E\end{array}\right) | | |
| 0902.0431_FO2478 | 148 | 0.998 | \varphi: \operatorname{Sp}(4) \rightarrow\left(E_{6}\right)^{\tau \gamma}, \varphi(A) X= | | |
| 0902.0431_FO2479 | 148 | 1.000 | g^{-1}\left(A(g X) A^{*}\right), X \in \mathfrak{J}^{C} | | |
| 0902.0431_FO2480 | 148 | 1.000 | \varphi_{*}: \mathfrak{s p}(4) \rightarrow\left(\mathfrak{e}_{6}\right)^{\tau \gamma}, \varphi_{*}(D) X= | | |
| 0902.0431_FO2481 | 148 | 1.000 | g^{-1}\left(D(g X)+(g X) D^{*}\right), X \in \mathfrak{J}^{C} | | |
| 0902.0431_FO2482 | 148 | 1.000 | \left(\mathfrak{e}_{7}\right)^{\tau \gamma} | | |
| 0902.0431_FO2483 | 148 | 1.000 | \boldsymbol{\mathfrak { S }}(8, \boldsymbol{C}) | | |
| 0902.0431_FO2484 | 148 | 1.000 | k_{J}: \mathfrak{J}(4, \boldsymbol{H})^{C} \rightarrow \mathfrak{S}(8, \boldsymbol{C})^{C} | | |
| 0902.0431_FO2485 | 148 | 0.999 | J=\operatorname{diag}(J, J, J, J) \in M(8, C), J=\left(\begin{array}{cc}0 & 1 \\ -1 & 0\end{array}\right) | | |
| 0902.0431_FO2486 | 148 | 0.658 | \chi: \mathfrak{P}^{C} \rightarrow \mathfrak{S}(8, \boldsymbol{C})^{C} | | |
| 0902.0431_FO2487 | 148 | 1.000 | \quad\left(\mathfrak{e}_{7}\right)^{\tau \gamma} \cong \mathfrak{s u}(8) | | |
| 0902.0431_FO2488 | 148 | 0.822 | \varphi_{*}: \mathfrak{s} \mathfrak{u}(8) \rightarrow\left(\mathfrak{e}_{7}\right)^{\tau \gamma} | | |
| 0902.0431_FO2489 | 149 | 1.000 | \varphi_{*}(B) \in\left(\mathfrak{e}_{7}\right)^{\tau \gamma} | | |
| 0902.0431_FO2490 | 149 | 1.000 | B=k(D), D \in \mathfrak{s p}(4) | | |
| 0902.0431_FO2491 | 149 | 0.815 | \varphi_{*}(C) X=g^{-1}\left(C(g X)+(g X) C^{*}\right) | | |
| 0902.0431_FO2492 | 149 | 1.000 | \varphi_{*}(k(D))=\Phi\left(\varphi_{*}(D), 0,0,0\right) \in\left(\mathfrak{e}_{7}\right)^{\tau \gamma} | | |
| 0902.0431_FO2493 | 149 | 0.995 | B=e_{1} k(T), T \in \mathfrak{J}(4, \boldsymbol{H})_{0}\left(\right. | | |
| 0902.0431_FO2494 | 149 | 0.995 | \left.T=g A, A \in\left(\mathfrak{J}^{C}\right)_{\tau \gamma}\right) | | |
| 0902.0431_FO2495 | 149 | 1.000 | \varphi_{*}\left(e_{1} k(T)\right)=\Phi(0, A,-\gamma A, 0) \in\left(\mathfrak{e}_{7}\right)^{\tau \gamma} | | |
| 0902.0431_FO2496 | 150 | 0.997 | \varphi: \mathfrak{s u}(8) \rightarrow\left(\mathfrak{e}_{7}\right)^{\tau \gamma} | | |
| 0902.0431_FO2497 | 150 | 0.998 | \quad\left(E_{7}\right)^{\tau \gamma} \cong S U(8) / \boldsymbol{Z}_{2}, \quad \boldsymbol{Z}_{2}=\{E,-E\} | | |
| 0902.0431_FO2498 | 150 | 1.000 | \varphi: S U(8) \rightarrow\left(E_{7}\right)^{\tau \gamma} | | |
| 0902.0431_FO2499 | 150 | 0.615 | \varphi(A) \in\left(E_{7}\right)^{\tau \gamma} | | |
| 0902.0431_FO2500 | 150 | 0.615 | \varphi_{*}: \mathfrak{s u}(8) \rightarrow | | |
| 0902.0431_FO2501 | 150 | 1.000 | \varphi_{*}(D) \in\left(\mathfrak{e}_{7}\right)^{\tau \gamma} | | |
| 0902.0431_FO2502 | 150 | 0.983 | \varphi_{*}: \mathfrak{s u}(8) \rightarrow\left(\mathfrak{e}_{7}\right)^{\tau \gamma} | | |
| 0902.0431_FO2503 | 150 | 1.000 | S U(8) / \boldsymbol{Z}_{2} \cong\left(E_{7}\right)^{\tau \gamma} | | |
| 0902.0431_FO2504 | 150 | 1.000 | \varphi: S U(8) \rightarrow | | |
| 0902.0431_FO2505 | 150 | 1.000 | a \in \boldsymbol{R}, \alpha_{i}(a) | | |
| 0902.0431_FO2506 | 150 | 1.000 | \varphi(S U(8)) | | |
| 0902.0431_FO2507 | 150 | 0.582 | \alpha_{i}(a)=\exp \left(\Phi\left(0,-a E_{i}, a E_{i}, 0\right)\right) \in \exp \varphi_{*}(\mathfrak{s u}(8)) | | |
| 0902.0431_FO2508 | 150 | 0.991 | \varphi(\exp (\mathfrak{s u}(8))) \in \varphi(S U(8)) | | |
| 0902.0431_FO2509 | 150 | 1.000 | P \in\left(\mathfrak{M}^{C}\right)_{\tau \gamma}=\left\{P \in \mathfrak{M}^{C} \mid \tau \gamma P=P\right\} | | |
| 0902.0431_FO2510 | 150 | 1.000 | \alpha \in \varphi(S U(8)) | | |
| 0902.0431_FO2511 | 150 | 1.000 | P=(X, Y, \xi, \eta) \in\left(\mathfrak{M}^{C}\right)_{\tau \gamma} | | |
| 0902.0431_FO2512 | 150 | 1.000 | Y \in\left(\mathfrak{J}^{C}\right)_{\tau \gamma} | | |
| 0902.0431_FO2513 | 150 | 1.000 | \gamma Y \in\left(\mathfrak{J}^{C}\right)_{\tau \gamma} | | |
| 0902.0431_FO2514 | 150 | 1.000 | g(\gamma Y) \in \mathfrak{J}(4, \boldsymbol{H})_{0} | | |
| 0902.0431_FO2515 | 150 | 0.998 | D \in \operatorname{Sp}(4) | | |
| 0902.0431_FO2516 | 151 | 1.000 | \xi<0 | | |
| 0902.0431_FO2517 | 151 | 1.000 | \alpha_{1}(\pi) | | |
| 0902.0431_FO2518 | 151 | 1.000 | \alpha \in\left(E_{7}\right)^{\tau \gamma} | | |
| 0902.0431_FO2519 | 151 | 1.000 | P=\alpha \mathrm{i} \in\left(\mathfrak{M}^{C}\right)_{\tau \gamma} | | |
| 0902.0431_FO2520 | 151 | 0.997 | \beta \in \varphi(S U(8)) | | |
| 0902.0431_FO2521 | 151 | 0.973 | a_{i}=\frac{\eta_{i}}{\left|\eta_{i}\right|} r_{i}\left(\eta_{i}\right. | | |
| 0902.0431_FO2522 | 151 | 0.973 | \eta_{i} \in \boldsymbol{R} | | |
| 0902.0431_FO2523 | 151 | 0.973 | \alpha_{i}\left(a_{i}\right) \in | | |
| 0902.0431_FO2524 | 151 | 0.997 | \widetilde{\alpha}=\alpha_{1}\left(a_{1}\right)^{-1} \alpha_{2}\left(a_{2}\right)^{-1} \alpha_{3}\left(a_{3}\right)^{-1} \beta \alpha | | |
| 0902.0431_FO2525 | 151 | 0.997 | \widetilde{\alpha} \in E_{6} | | |
| 0902.0431_FO2526 | 151 | 0.995 | \tau \gamma \widetilde{\alpha}=\widetilde{\alpha} \tau \gamma | | |
| 0902.0431_FO2527 | 151 | 0.995 | \widetilde{\alpha} \in\left(E_{6}\right)^{\tau \gamma}=\varphi(S p(4)) | | |
| 0902.0431_FO2528 | 151 | 0.997 | \subset \varphi(S U(8)) | | |
| 0902.0431_FO2529 | 151 | 0.997 | \alpha=\beta^{-1} \alpha_{3}\left(a_{3}\right) \alpha_{2}\left(a_{2}\right) \alpha_{1}\left(a_{1}\right) \widetilde{\alpha} \in \varphi(S U(8)) | | |
| 0902.0431_FO2530 | 151 | 1.000 | w \in G_{2} \subset F_{4} \subset E_{6} \subset E_{7} | | |
| 0902.0431_FO2531 | 151 | 1.000 | \left(E_{7}\right)^{w} | | |
| 0902.0431_FO2532 | 151 | 1.000 | E_{7, \boldsymbol{C}} | | |
| 0902.0431_FO2533 | 152 | 0.999 | \left((S U(3) \times S U(3)) / \boldsymbol{Z}_{3}\right) \cdot \boldsymbol{Z}_{2} | | |
| 0902.0431_FO2534 | 152 | 1.000 | \mathfrak{e}_{7, \boldsymbol{C}} | | |
| 0902.0431_FO2535 | 152 | 1.000 | \pi_{0}\left(E_{6, \boldsymbol{C}}\right) \rightarrow \pi_{0}\left(E_{7, \boldsymbol{C}}\right) \rightarrow \pi_{0}\left(\left(\mathfrak{M}_{\boldsymbol{C}}\right)_{1}\right) | | |
| 0902.0431_FO2536 | 152 | 1.000 | \boldsymbol{Z}_{2} \rightarrow \pi_{0}\left(E_{7 . \boldsymbol{C}}\right) \rightarrow 0 | | |
| 0902.0431_FO2537 | 152 | 1.000 | \pi_{0}\left(E_{7, \boldsymbol{C}}\right) | | |
| 0902.0431_FO2538 | 152 | 0.989 | h^{\prime}: C \rightarrow \boldsymbol{C} | | |
| 0902.0431_FO2539 | 152 | 0.996 | V, W | | |
| 0902.0431_FO2540 | 152 | 0.996 | f: V \rightarrow | | |
| 0902.0431_FO2541 | 152 | 0.415 | \boldsymbol{C}-\boldsymbol{C} | | |
| 0902.0431_FO2542 | 152 | 0.415 | g: W \rightarrow V | | |
| 0902.0431_FO2543 | 152 | 0.886 | h^{\prime}: \boldsymbol{C}^{C} \rightarrow \boldsymbol{C} | | |
| 0902.0431_FO2544 | 152 | 0.886 | C-\boldsymbol{C} | | |
| 0902.0431_FO2545 | 152 | 0.999 | \Lambda^{3}\left(\boldsymbol{C}^{6}\right) | | |
| 0902.0431_FO2546 | 152 | 0.999 | \boldsymbol{C}^{6} | | |
| 0902.0431_FO2547 | 152 | 0.818 | f:\left(\mathfrak{P}_{\boldsymbol{C}}\right)^{C} \rightarrow \Lambda^{3}\left(\boldsymbol{C}^{6}\right) | | |
| 0902.0431_FO2548 | 152 | 1.000 | \left(\left\{\boldsymbol{e}_{1}, \boldsymbol{e}_{2}, \cdots, \boldsymbol{e}_{6}\right\}\right. | | |
| 0902.0431_FO2549 | 152 | 1.000 | x_{i j k} \in \boldsymbol{C} | | |
| 0902.0431_FO2550 | 152 | 0.984 | \left.x_{i^{\prime} j^{\prime} k^{\prime}}=\operatorname{sgn}\left(\begin{array}{ccc}i & j & k \\ i^{\prime} & j^{\prime} & k^{\prime}\end{array}\right) x_{i j k}\right) | | |
| 0902.0431_FO2551 | 153 | 1.000 | f^{-1}: \Lambda^{3}\left(\boldsymbol{C}^{6}\right) \rightarrow\left(\mathfrak{P}_{\boldsymbol{C}}\right)^{C} | | |
| 0902.0431_FO2552 | 153 | 0.979 | h: \boldsymbol{C} \oplus \boldsymbol{C} \rightarrow \boldsymbol{C}^{C}, h: \boldsymbol{C} \rightarrow C | | |
| 0902.0431_FO2553 | 153 | 1.000 | S U(6) | | |
| 0902.0431_FO2554 | 153 | 1.000 | \boldsymbol{a} \wedge \boldsymbol{b} \wedge \boldsymbol{c} \in \Lambda^{3}\left(\boldsymbol{C}^{6}\right) | | |
| 0902.0431_FO2555 | 153 | 1.000 | D \in \mathfrak{s u}(6) | | |
| 0902.0431_FO2556 | 154 | 1.000 | \mathfrak{s u}(6) | | |
| 0902.0431_FO2557 | 154 | 0.342 | \varphi_{\boldsymbol{C}}: \mathfrak{s u}(6) \rightarrow_{\mathfrak{e}_{7, \boldsymbol{C}}} | | |
| 0902.0431_FO2558 | 154 | 1.000 | \phi_{\boldsymbol{C}}(B, C) X=h(B, C) X+X h(B, C)^{*} | | |
| 0902.0431_FO2559 | 154 | 1.000 | X \in\left(\mathfrak{J}_{\boldsymbol{C}}\right)^{C} | | |
| 0902.0431_FO2560 | 154 | 0.980 | D=\left(\begin{array}{cc}B^{\prime} & L \\ -L^{*} & C^{\prime}\end{array}\right) \in \mathfrak{s u}(6), B^{\prime}, C^{\prime} \in \mathfrak{u}(3), L \in M(3, \boldsymbol{C}) | | |
| 0902.0431_FO2561 | 154 | 1.000 | E, A_{i j} \in M(3, \boldsymbol{C}) | | |
| 0902.0431_FO2562 | 154 | 1.000 | S U(6) \cdot \boldsymbol{Z}_{2} | | |
| 0902.0431_FO2563 | 154 | 0.977 | E_{7, \boldsymbol{C}} \cong\left(S U(6) / \boldsymbol{Z}_{3}\right) \cdot \boldsymbol{Z}_{2}, \boldsymbol{Z}_{3}=\left\{E, \omega_{1} E, \omega_{1}{ }^{2} E\right\}, \omega_{1}= | | |
| 0902.0431_FO2564 | 154 | 1.000 | \psi: S U(6) \cdot \boldsymbol{Z}_{2} \rightarrow E_{7, \boldsymbol{C}} | | |
| 0902.0431_FO2565 | 154 | 1.000 | \psi(A, 1) \in E_{7, \boldsymbol{C}} | | |
| 0902.0431_FO2566 | 154 | 1.000 | \psi_{*}: \mathfrak{s u}(6) \rightarrow \mathfrak{e}_{7, \boldsymbol{C}} | | |
| 0902.0431_FO2567 | 154 | 1.000 | \psi | | |
| 0902.0431_FO2568 | 154 | 0.998 | \psi_{\boldsymbol{C}}: \mathfrak{s u}(6) \rightarrow \mathfrak{e}_{7, \boldsymbol{C}} | | |
| 0902.0431_FO2569 | 155 | 1.000 | \bar{b}_{i i}=-b_{i i}, \bar{c}_{i i}=-c_{i i}, b_{11}+b_{22}+b_{33}=c_{11}+c_{22}+c_{33}=0, \bar{\nu}=-\nu | | |
| 0902.0431_FO2570 | 155 | 1.000 | P=(0,0,1,0), \psi_{*}(D) P | | |
| 0902.0431_FO2571 | 155 | 1.000 | P=\left(E_{1}, 0,0,0\right), \psi_{*}(D) P | | |
| 0902.0431_FO2572 | 156 | 1.000 | P=\left(F_{1}(1), 0,0,0\right), \psi_{*}(D) P | | |
| 0902.0431_FO2573 | 157 | 0.977 | \mathfrak{P}_{\boldsymbol{C}}{ }^{C} | | |
| 0902.0431_FO2574 | 157 | 0.977 | P=(X, 0,0,0),(0, X, 0,0) | | |
| 0902.0431_FO2575 | 157 | 1.000 | X=E_{i}, F_{i}(1), F_{i}\left(e_{1}\right), i=1,2,3 | | |
| 0902.0431_FO2576 | 157 | 1.000 | P=(0,0,0,1) | | |
| 0902.0431_FO2577 | 157 | 0.999 | \psi(E, \epsilon) P=\bar{P} | | |
| 0902.0431_FO2578 | 157 | 1.000 | \psi(E, \epsilon)=\epsilon \in G_{2, \boldsymbol{C}}(=\operatorname{Aut}(\boldsymbol{C})) \subset F_{4, \boldsymbol{C}} \subset E_{6, \boldsymbol{C}} \subset E_{7, \boldsymbol{C}} | | |
| 0902.0431_FO2579 | 157 | 0.997 | \psi(A, \epsilon)=\varphi(A, 1) \varphi(E, \epsilon) \in E_{7, \boldsymbol{C}} | | |
| 0902.0431_FO2580 | 157 | 0.789 | \psi_{*}: \mathfrak{s u}(6) \rightarrow \mathfrak{e}_{7, \boldsymbol{C}}, \psi: S U(6) \rightarrow\left(E_{7, \boldsymbol{C}}\right)_{0} | | |
| 0902.0431_FO2581 | 157 | 0.997 | \epsilon=\psi(E, \epsilon) \notin\left(E_{7, \boldsymbol{C}}\right)_{0} | | |
| 0902.0431_FO2582 | 157 | 0.817 | E_{7, C} | | |
| 0902.0431_FO2583 | 157 | 1.000 | \operatorname{Ker} \psi=\left\{E, \omega_{1} E, \omega_{1}{ }^{2} E\right\} \times 1=\boldsymbol{Z}_{3} \times 1 | | |
| 0902.0431_FO2584 | 157 | 1.000 | \left(S U(6) / \boldsymbol{Z}_{3}\right) \cdot \boldsymbol{Z}_{2} \cong E_{7, \boldsymbol{C}} | | |
| 0902.0431_FO2585 | 157 | 1.000 | \left(\mathfrak{P}_{\boldsymbol{C}}\right)^{C} \oplus\left(M(3, \boldsymbol{C})^{C} \oplus M(3, \boldsymbol{C})^{C}\right) | | |
| 0902.0431_FO2586 | 157 | 1.000 | \mathfrak{J}_{\boldsymbol{C}}{ }^{C} \oplus M(3, \boldsymbol{C})^{C} | | |
| 0902.0431_FO2587 | 158 | 0.822 | \mu: M(6, \boldsymbol{C}) \rightarrow M(3, \boldsymbol{C})^{C} \oplus M(3, \boldsymbol{C})^{C} | | |
| 0902.0431_FO2588 | 158 | 1.000 | M_{i j} \in M(3, \boldsymbol{C}) | | |
| 0902.0431_FO2589 | 158 | 1.000 | \mu^{-1}: M(3, \boldsymbol{C})^{C} \oplus M(3, \boldsymbol{C})^{C} \rightarrow M(6, \boldsymbol{C}) | | |
| 0902.0431_FO2590 | 158 | 1.000 | \widetilde{M} \in M(6, \boldsymbol{C}) | | |
| 0902.0431_FO2591 | 159 | 0.615 | \phi_{\boldsymbol{C}}(B, C) M=M \tau h(B, C)^{*}=-M h(B, C) | | |
| 0902.0431_FO2592 | 159 | 0.615 | -2 \tau h(L) \times N=N \tau h(L) | | |
| 0902.0431_FO2593 | 159 | 0.797 | \xrightarrow{\mu^{-1}} \cdots | | |
| 0902.0431_FO2594 | 159 | 0.797 | \cdots | | |
| 0902.0431_FO2595 | 159 | 0.809 | f: \mathfrak{P}^{C} \rightarrow \Lambda^{3}\left(\boldsymbol{C}^{6}\right) \oplus M(6, \boldsymbol{C}) | | |
| 0902.0431_FO2596 | 159 | 1.000 | S U(3) \times S U(6) | | |
| 0902.0431_FO2597 | 159 | 1.000 | \Lambda^{3}\left(\boldsymbol{C}^{6}\right) \oplus M(6, \boldsymbol{C}) | | |
| 0902.0431_FO2598 | 159 | 1.000 | Q \widetilde{M} | | |
| 0902.0431_FO2599 | 159 | 1.000 | \left(\begin{array}{cc}Q & 0 \\ 0 & Q\end{array}\right)\left(\begin{array}{ll}M_{11} & M_{12} \\ M_{21} & M_{22}\end{array}\right)=\left(\begin{array}{ll}Q M_{11} & Q M_{12} \\ Q M_{21} & Q M_{22}\end{array}\right), M_{i j} \in M(3, \boldsymbol{C}) | | |
| 0902.0431_FO2600 | 159 | 0.918 | \left(E_{7}\right)^{w} \cong(S U(3) \times S U(6)) / \boldsymbol{Z}_{3}, \quad \boldsymbol{Z}_{3}=\left\{(E, E),\left(\omega_{1} E, \omega_{1} E\right)\right. | | |
| 0902.0431_FO2601 | 159 | 1.000 | \psi: S U(3) \times S U(6) \rightarrow\left(E_{7}\right)^{w} | | |
| 0902.0431_FO2602 | 159 | 1.000 | \psi(Q, A) \in E_{7} | | |
| 0902.0431_FO2603 | 159 | 1.000 | \varphi(Q, E) \in\left(E_{6}\right)^{w} \subset | | |
| 0902.0431_FO2604 | 159 | 1.000 | \psi(E, A) \in E_{7} | | |
| 0902.0431_FO2605 | 159 | 0.910 | \psi_{*}: \mathfrak{s u}(3) \oplus \mathfrak{s u}(6) \rightarrow \mathfrak{e}_{7} | | |
| 0902.0431_FO2606 | 159 | 0.910 | \psi, \psi_{*}(0, D) | | |
| 0902.0431_FO2607 | 159 | 1.000 | \left(\mathfrak{P}^{C}\right)_{w}=\left\{P \in \mathfrak{P}^{C} \mid w P=P\right\}=\left(\mathfrak{P}_{\boldsymbol{C}}\right)^{C} | | |
| 0902.0431_FO2608 | 159 | 1.000 | w \psi(Q, A)=\psi(Q, A) w | | |
| 0902.0431_FO2609 | 159 | 1.000 | \psi(Q, A) \in\left(E_{7}\right)^{w} | | |
| 0902.0431_FO2610 | 159 | 1.000 | \alpha \in\left(E_{7}\right)^{w} | | |
| 0902.0431_FO2611 | 159 | 1.000 | \left(\mathfrak{P}^{C}\right)_{w}=\left(\mathfrak{P}_{\boldsymbol{C}}\right)^{C} | | |
| 0902.0431_FO2612 | 159 | 0.999 | \beta=\psi(E, A)^{-1} \alpha | | |
| 0902.0431_FO2613 | 159 | 0.999 | \beta \mid\left(\mathfrak{P}_{\boldsymbol{C}}\right)^{C}=1 | | |
| 0902.0431_FO2614 | 159 | 0.999 | \beta \in\left(G_{2}\right)^{w}=S U(3) | | |
| 0902.0431_FO2615 | 159 | 1.000 | Q \in S U(3) | | |
| 0902.0431_FO2616 | 160 | 1.000 | \gamma_{1} \in G_{2} \subset F_{4} \subset E_{6} \subset E_{7} | | |
| 0902.0431_FO2617 | 160 | 1.000 | \gamma_{1} \alpha \in | | |
| 0902.0431_FO2618 | 160 | 0.998 | \gamma_{1} \in\left(G_{2}\right)^{w}=S U(3) | | |
| 0902.0431_FO2619 | 160 | 1.000 | \operatorname{Ker} \psi=\left\{(E, E),\left(\omega_{1} E, \omega_{1} E\right),\left(\omega_{1}{ }^{2} E, \omega_{1}{ }^{2} E\right)\right\}=\boldsymbol{Z}_{3} | | |
| 0902.0431_FO2620 | 160 | 0.999 | (S U(3) \times S U(6)) / \boldsymbol{Z}_{3} \cong\left(E_{7}\right)^{w} | | |
| 0902.0431_FO2621 | 160 | 1.000 | \left((S U(3) \times S U(6)) / \boldsymbol{Z}_{3}\right) \cdot \boldsymbol{Z}_{2} | | |
| 0902.0431_FO2622 | 160 | 1.000 | \boldsymbol{Z}_{2}=\{1, \gamma\} | | |
| 0902.0431_FO2623 | 160 | 0.953 | \left.\gamma(Q, A)=\left(\bar{Q}, \overline{\operatorname{Ad}\left(J_{3}\right) A}\right)\right) | | |
| 0902.0431_FO2624 | 160 | 1.000 | \left(\mathfrak{e}_{7}\right)^{w} | | |
| 0902.0431_FO2625 | 160 | 1.000 | \operatorname{dim}\left(\mathfrak{e}_{7}\right)^{w}=10+14+6 \times 3+1=43=8+35= | | |
| 0902.0431_FO2626 | 160 | 0.995 | \operatorname{dim}(\mathfrak{s u}(3) \oplus \mathfrak{s u}(6)) | | |
| 0902.0431_FO2627 | 160 | 1.000 | \operatorname{Iso}_{C}\left(\mathfrak{P}^{C}\right)=G L(78, C) | | |
| 0902.0431_FO2628 | 160 | 1.000 | \langle X, Y\rangle:\langle\alpha X, Y\rangle=\left\langle X, \alpha^{*} Y\right\rangle | | |
| 0902.0431_FO2629 | 160 | 1.000 | \alpha^{*}=\tau \lambda \alpha^{-1} \lambda^{-1} \tau \in E_{7}{ }^{C} | | |
| 0902.0431_FO2630 | 161 | 0.993 | P, Q \in \mathfrak{P} | | |
| 0902.0431_FO2631 | 161 | 0.993 | \mathfrak{P}^{\prime} | | |
| 0902.0431_FO2632 | 161 | 0.993 | P \times Q: \mathfrak{P} \rightarrow \mathfrak{P} | | |
| 0902.0431_FO2633 | 161 | 0.993 | \mathfrak{P}^{\prime} \rightarrow \mathfrak{P}^{\prime} | | |
| 0902.0431_FO2634 | 161 | 1.000 | \langle P, Q\rangle_{\sigma} | | |
| 0902.0431_FO2635 | 162 | 1.000 | 133+56 \times 2+3=248 | | |
| 0902.0431_FO2636 | 162 | 1.000 | \left[R_{1}, R_{2}\right] | | |
| 0902.0431_FO2637 | 163 | 1.000 | \mathfrak{K}^{C}=\mathfrak{P}^{C} \oplus \mathfrak{P}^{C} \oplus C \oplus C \oplus C | | |
| 0902.0431_FO2638 | 163 | 1.000 | p: \mathfrak{e}_{8}{ }^{C} \rightarrow \mathfrak{e}_{7}{ }^{C} | | |
| 0902.0431_FO2639 | 163 | 1.000 | q: \mathfrak{e}_{8}{ }^{C} \rightarrow \mathfrak{K}^{C} | | |
| 0902.0431_FO2640 | 163 | 1.000 | \mathfrak{e}_{8}{ }^{C}=\mathfrak{e}_{7}{ }^{C} \oplus \mathfrak{K}^{C} | | |
| 0902.0431_FO2641 | 163 | 1.000 | \Phi \in p(\mathfrak{a}) | | |
| 0902.0431_FO2642 | 163 | 1.000 | (0, P, Q, r, s, t) \in \mathfrak{K}^{C} | | |
| 0902.0431_FO2643 | 163 | 0.970 | (\Phi, P, Q, r, s, t) \in \mathfrak{a} | | |
| 0902.0431_FO2644 | 163 | 0.970 | \Phi_{1} \in \mathfrak{e}_{7}{ }^{C} | | |
| 0902.0431_FO2645 | 163 | 1.000 | \left[\Phi_{1}, \Phi\right] \in p(\mathfrak{a}) | | |
| 0902.0431_FO2646 | 163 | 1.000 | \mathfrak{e}_{7}{ }^{C} \cap \mathfrak{a} \neq\{0\} | | |
| 0902.0431_FO2647 | 163 | 1.000 | \mathfrak{K}^{C} \cap \mathfrak{a} \neq\{0\} | | |
| 0902.0431_FO2648 | 163 | 1.000 | \mathfrak{e}_{7}{ }^{C} \cap \mathfrak{a}=\{0\} | | |
| 0902.0431_FO2649 | 163 | 1.000 | \mathfrak{K}^{C} \cap \mathfrak{a}=\{0\} | | |
| 0902.0431_FO2650 | 163 | 1.000 | p \mid \mathfrak{a}: \mathfrak{a} \rightarrow \mathfrak{e}_{7}{ }^{C} | | |
| 0902.0431_FO2651 | 163 | 1.000 | p(\mathfrak{a})=\mathfrak{e}_{7}{ }^{C} | | |
| 0902.0431_FO2652 | 163 | 1.000 | \operatorname{dim}_{C}(\mathfrak{a})=\operatorname{dim}_{C}(p(\mathfrak{a}))=\operatorname{dim}_{C}\left(\mathfrak{e}_{7}{ }^{C}\right)=133 | | |
| 0902.0431_FO2653 | 163 | 1.000 | q \mid \mathfrak{a}: \mathfrak{a} \rightarrow \mathfrak{K}^{C} | | |
| 0902.0431_FO2654 | 163 | 1.000 | \operatorname{dim}_{C}(\mathfrak{a}) \leq \operatorname{dim}_{C}\left(\mathfrak{K}^{C}\right)=56 \times 2+3=115 | | |
| 0902.0431_FO2655 | 163 | 1.000 | \mathfrak{e}_{7}{ }^{C} \cap \mathfrak{a}=\mathfrak{e}_{7}{ }^{C} | | |
| 0902.0431_FO2656 | 163 | 1.000 | \mathfrak{a} \supset \mathfrak{e}_{7}{ }^{C} | | |
| 0902.0431_FO2657 | 163 | 1.000 | \mathfrak{a} \supset \mathfrak{e}_{7}{ }^{C} \oplus \mathfrak{K}^{C}=\mathfrak{e}_{8}{ }^{C} | | |
| 0902.0431_FO2658 | 163 | 1.000 | \mathfrak{a}=\mathfrak{e}_{8}{ }^{C} | | |
| 0902.0431_FO2659 | 163 | 1.000 | R=(0, P, Q, r, s, t) | | |
| 0902.0431_FO2660 | 163 | 1.000 | \mathfrak{K}^{C} \cap \mathfrak{a} \subset \mathfrak{a} | | |
| 0902.0431_FO2661 | 163 | 1.000 | R=(0, P, Q, r, s, t), P \neq 0 | | |
| 0902.0431_FO2662 | 163 | 0.996 | P_{1} \in \mathfrak{P}^{C} | | |
| 0902.0431_FO2663 | 163 | 0.996 | P \times P_{1} \neq 0 | | |
| 0902.0431_FO2664 | 163 | 0.911 | \left[\Phi, P \times P_{1}\right] \neq 0 | | |
| 0902.0431_FO2665 | 164 | 1.000 | R=(0, P, Q, r, s, t), Q \neq 0 | | |
| 0902.0431_FO2666 | 164 | 1.000 | R=(0,0,0, r, s, t), r \neq 0 | | |
| 0902.0431_FO2667 | 164 | 1.000 | 0 \neq P \in \mathfrak{P}^{C} | | |
| 0902.0431_FO2668 | 164 | 1.000 | R=(0,0,0,0, s, t), s \neq 0 | | |
| 0902.0431_FO2669 | 164 | 1.000 | R=(0,0,0,0,0, t), t \neq 0 | | |
| 0902.0431_FO2670 | 164 | 0.942 | R \in \mathfrak{e}_{8}{ }^{C} | | |
| 0902.0431_FO2671 | 164 | 0.942 | \operatorname{ad} R: \mathfrak{e}_{8}{ }^{C} \rightarrow \mathfrak{e}_{8}{ }^{C} | | |
| 0902.0431_FO2672 | 164 | 0.942 | \operatorname{ad} R\left(R_{1}\right)=\left[R, R_{1}\right] | | |
| 0902.0431_FO2673 | 164 | 0.942 | \Theta(R)= | | |
| 0902.0431_FO2674 | 164 | 0.999 | \operatorname{ad} R | | |
| 0902.0431_FO2675 | 164 | 0.996 | \Theta(R) | | |
| 0902.0431_FO2676 | 164 | 0.996 | R | | |
| 0902.0431_FO2677 | 164 | 0.993 | \mathfrak{e}_{8}{ }^{C} \cong \operatorname{Der}\left(\mathfrak{e}_{8}{ }^{C}\right) | | |
| 0902.0431_FO2678 | 164 | 0.994 | \left(R_{1}, R_{2}\right)_{8} | | |
| 0902.0431_FO2679 | 164 | 0.766 | R_{i}=\left(\Phi_{i}, P_{i}, Q_{i}, r_{i}, s_{i}, t_{i}\right) \in \mathfrak{e}_{8}{ }^{C} | | |
| 0902.0431_FO2680 | 164 | 1.000 | \left(\left[R, R_{1}\right], R_{2}\right)_{8} | | |
| 0902.0431_FO2681 | 165 | 1.000 | B_{8} | | |
| 0902.0431_FO2682 | 165 | 0.737 | R_{i}=\left(\Phi_{i}, P_{i}, Q_{i}, r_{i}, s_{i}, t_{i}\right) \in \mathfrak{e}_{8}^{C} | | |
| 0902.0431_FO2683 | 165 | 1.000 | R_{1}=R_{2}=(0,0,0,1,0,0)=1 | | |
| 0902.0431_FO2684 | 165 | 1.000 | k=-15 | | |
| 0902.0431_FO2685 | 165 | 1.000 | B_{8}\left(R_{1}, R_{2}\right)=-15\left(R_{1}, R_{2}\right)_{8} | | |
| 0902.0431_FO2686 | 166 | 1.000 | \operatorname{Inn}\left(\mathfrak{e}_{8}{ }^{C}\right) | | |
| 0902.0431_FO2687 | 166 | 1.000 | \exp (\Theta(R)), R \in \mathfrak{e}_{8}{ }^{C} | | |
| 0902.0431_FO2688 | 166 | 1.000 | \operatorname{Aut}\left(\mathfrak{e}_{8}{ }^{C}\right)=E_{8}{ }^{C} | | |
| 0902.0431_FO2689 | 166 | 0.988 | \operatorname{Aut}\left(\mathfrak{e}_{8}{ }^{C}\right) / \operatorname{Inn}\left(\mathfrak{e}_{8}{ }^{C}\right)=\{1\} | | |
| 0902.0431_FO2690 | 166 | 1.000 | R \times R: \mathfrak{e}_{8}{ }^{C} \rightarrow \mathfrak{e}_{8}{ }^{C} | | |
| 0902.0431_FO2691 | 166 | 1.000 | \mathfrak{W}^{C} | | |
| 0902.0431_FO2692 | 166 | 0.999 | \left(E_{8}{ }^{C}\right)_{1_{-}}=\left\{\alpha \in E_{8}{ }^{C} \mid \alpha 1_{-}=1_{-}\right\}=\exp \left(\Phi\left(0,0, \mathfrak{P}^{C}, 0,0, C\right)\right) E_{7} | | |
| 0902.0431_FO2693 | 166 | 0.997 | \left(E_{8}{ }^{C}\right)_{1_{-}} | | |
| 0902.0431_FO2694 | 166 | 1.000 | \operatorname{Der}\left(\mathfrak{e}_{8}{ }^{C}\right) \cong \mathfrak{e}_{8}{ }^{C} | | |
| 0902.0431_FO2695 | 166 | 1.000 | \lambda, \lambda^{\prime} | | |
| 0902.0431_FO2696 | 166 | 0.973 | \lambda^{\prime} | | |
| 0902.0431_FO2697 | 166 | 0.973 | \lambda, \lambda^{\prime} \in \operatorname{Aut}\left(\mathfrak{e}_{8}{ }^{C}\right)=E_{8}{ }^{C} | | |
| 0902.0431_FO2698 | 167 | 0.994 | \left\langle R_{1}, R_{2}\right\rangle | | |
| 0902.0431_FO2699 | 167 | 0.772 | R_{i}=\left(\Phi_{i}, P_{i}, Q_{i}, r_{i}, s_{i}, t_{i}\right) \in \mathfrak{e}_{8}{ }^{C}, i=1,2 | | |
| 0902.0431_FO2700 | 167 | 0.772 | \tau \widetilde{\lambda} R_{1}=\left(\tau \lambda \Phi_{1} \lambda^{-1} \tau\right. | | |
| 0902.0431_FO2701 | 167 | 0.853 | \tau \lambda Q_{1},-\tau \lambda P_{1},-\tau r_{1},-\tau t_{1},-\tau s_{1} | | |
| 0902.0431_FO2702 | 167 | 1.000 | \left(\tau \lambda \Phi_{1} \lambda^{-1} \tau, \Phi_{2}\right)_{7} | | |
| 0902.0431_FO2703 | 167 | 1.000 | \Phi_{i}= | | |
| 0902.0431_FO2704 | 167 | 1.000 | \Phi\left(\phi_{i}, A_{i}, B_{i}, \nu_{i}\right), i=1,2 | | |
| 0902.0431_FO2705 | 167 | 1.000 | \tau \lambda \Phi_{1} \lambda^{-1} \tau=\Phi\left(-\tau^{t} \phi_{1} \tau,-\tau B_{1},-\tau A_{1},-\tau \nu_{1}\right) | | |
| 0902.0431_FO2706 | 167 | 1.000 | \left(\tau^{t} \phi_{1} \tau, \phi_{2}\right)_{6} | | |
| 0902.0431_FO2707 | 167 | 1.000 | \phi_{i}=\delta_{i}+\widetilde{T}_{i} \in | | |
| 0902.0431_FO2708 | 167 | 0.996 | \mathfrak{e}_{6}{ }^{C}, \delta_{i} \in \mathfrak{f}_{4}{ }^{C}, \widetilde{T}_{i} \in \mathfrak{J}_{0}{ }^{C}, i=1,2 | | |
| 0902.0431_FO2709 | 167 | 0.996 | \tau^{t} \phi_{1} \tau=-\tau \delta_{1} \tau+\tau \widetilde{T}_{1} | | |
| 0902.0431_FO2710 | 167 | 1.000 | -\left(\tau \delta_{1} \tau, \delta_{2}\right)_{4} | | |
| 0902.0431_FO2711 | 168 | 1.000 | \mathfrak{e}_{8} | | |
| 0902.0431_FO2712 | 168 | 1.000 | R=(\Phi, P, Q, r, s, t) \in \mathfrak{e}_{8}{ }^{C} | | |
| 0902.0431_FO2713 | 168 | 1.000 | \tau \widetilde{\lambda} R=R | | |
| 0902.0431_FO2714 | 168 | 1.000 | \tau \lambda \Phi=\Phi \lambda \tau, Q=-\tau \lambda P, \tau r=-r, t=-\tau s | | |
| 0902.0431_FO2715 | 168 | 1.000 | R^{*} | | |
| 0902.0431_FO2716 | 168 | 0.975 | R^{*}=\tau \widetilde{\lambda} R \widetilde{\lambda} \tau \in \mathfrak{e}_{8}{ }^{C} | | |
| 0902.0431_FO2717 | 168 | 0.975 | R \in \mathfrak{e}_{8}{ }^{C}, R | | |
| 0902.0431_FO2718 | 168 | 1.000 | R^{*}=-R | | |
| 0902.0431_FO2719 | 168 | 1.000 | R \in \mathfrak{e}_{8} | | |
| 0902.0431_FO2720 | 168 | 1.000 | \mathfrak{e}_{8} \cong \Theta\left(\mathfrak{e}_{8}\right) | | |
| 0902.0431_FO2721 | 168 | 1.000 | \operatorname{Iso}_{C}\left(\mathfrak{e}_{8}{ }^{C}\right)=G L(248, C) | | |
| 0902.0431_FO2722 | 168 | 1.000 | \alpha \in E_{8}{ }^{C} | | |
| 0902.0431_FO2723 | 168 | 1.000 | \alpha^{*}=\tau \widetilde{\lambda} \alpha^{-1} \widetilde{\lambda} \tau \in E_{8}{ }^{C} | | |
| 0902.0431_FO2724 | 170 | 0.905 | \mathfrak{h}_{7} | | |
| 0902.0431_FO2725 | 170 | 1.000 | Y=0, \xi=\eta=0 | | |
| 0902.0431_FO2726 | 170 | 1.000 | X=F_{2}(a), F_{3}(a) | | |
| 0902.0431_FO2727 | 170 | 1.000 | X=0, \xi=\eta=0 | | |
| 0902.0431_FO2728 | 170 | 1.000 | Y=E_{k}, Y=F_{i}(a) | | |
| 0902.0431_FO2729 | 170 | 1.000 | X=Y=0, \xi=1, \eta=0 | | |
| 0902.0431_FO2730 | 170 | 1.000 | \nu+r | | |
| 0902.0431_FO2731 | 170 | 1.000 | -\nu+r | | |
| 0902.0431_FO2732 | 171 | 1.000 | 2 r | | |
| 0902.0431_FO2733 | 171 | 1.000 | -2 r | | |
| 0902.0431_FO2734 | 171 | 1.000 | n_{1} n_{2} \cdots n_{8} | | |
| 0902.0431_FO2735 | 171 | 0.998 | \cdots+n_{8} \alpha_{8} | | |
| 0902.0431_FO2765 | 176 | 0.998 | \Pi=\left\{\alpha_{1}, \alpha_{2}, \cdots, \alpha_{8}\right\} | | |
| 0902.0431_FO2766 | 176 | 0.925 | \Phi(h)=\left\{\Phi\left(\sum_{k=0}^{3} \lambda_{k} H_{k}+\left(\sum_{j=1}^{3} \mu_{j} E_{j}\right)^{\sim}, 0,0, \nu\right) \in\left(\mathfrak{h}_{7}\right)_{\boldsymbol{R}}\right. | | |
| 0902.0431_FO2767 | 176 | 1.000 | \widetilde{h}=(\Phi(h), 0,0, r, 0), \widetilde{h}^{\prime}=\left(\Phi\left(h^{\prime}\right), 0,0, r^{\prime}, 0\right) \in \mathfrak{h} | | |
| 0902.0431_FO2768 | 176 | 1.000 | \Phi(h)=\Phi\left(\sum_{k=0}^{3} \lambda_{k} H_{k}+\right. | | |
| 0902.0431_FO2769 | 176 | 0.449 | \left.\left(\sum_{j=1}^{3} \mu_{j} E_{j}\right)^{\sim}, 0,0, \nu\right), \Phi\left(h^{\prime}\right)=\Phi\left(\sum_{k=0}^{3} \lambda_{k}^{\prime} H_{k}+\left(\sum_{j=1}^{3} \mu_{j}^{\prime} E_{j}\right)^{\sim}, 0,0, \nu^{\prime}\right) \in\left(\mathfrak{h}_{7}\right)_{\boldsymbol{R}} | | |
| 0902.0431_FO2770 | 176 | 0.991 | \alpha_{i}\left(B_{8}\left(H_{\alpha}, H\right)=\alpha(H), H \in\right. | | |
| 0902.0431_FO2771 | 177 | 0.997 | \alpha_{1}, \alpha_{2}, \cdots, \alpha_{8} | | |
| 0902.0431_FO2772 | 177 | 1.000 | A_{1} \oplus E_{7} | | |
| 0902.0431_FO2773 | 177 | 1.000 | D_{8} | | |
| 0902.0431_FO2774 | 177 | 1.000 | A_{2} \oplus E_{6} | | |
| 0902.0431_FO2775 | 177 | 1.000 | A_{8} | | |
| 0902.0431_FO2776 | 177 | 1.000 | A_{4} \oplus A_{4} | | |
| 0902.0431_FO2777 | 178 | 0.999 | \left(E_{8}{ }^{C}\right)_{1,1^{-}, 1_{-}} | | |
| 0902.0431_FO2778 | 178 | 0.683 | \quad\left(E_{8}{ }^{C}\right)_{1,1^{-}, 1_{-}} \cong E_{7}{ }^{C} | | |
| 0902.0431_FO2779 | 178 | 1.000 | \beta \in E_{7}{ }^{C} | | |
| 0902.0431_FO2780 | 178 | 1.000 | \widetilde{\beta}: \mathfrak{e}_{8}{ }^{C} \rightarrow \mathfrak{e}_{8}{ }^{C} | | |
| 0902.0431_FO2781 | 178 | 0.992 | (\operatorname{Ad} \beta) \Phi=\beta \Phi \beta^{-1}, \Phi \in \mathfrak{e}_{7}{ }^{C} | | |
| 0902.0431_FO2782 | 178 | 0.992 | \widetilde{\beta} \in\left(E_{8}{ }^{C}\right)_{1,1^{-}, 1_{-}} | | |
| 0902.0431_FO2783 | 178 | 1.000 | \alpha 1=1, \alpha 1^{-}=1^{-} | | |
| 0902.0431_FO2784 | 178 | 1.000 | \alpha 1_{-}=1_{-} | | |
| 0902.0431_FO2785 | 178 | 1.000 | [\alpha \Phi, 1]=[\alpha \Phi, \alpha 1]=\alpha[\Phi, 1]=0 | | |
| 0902.0431_FO2786 | 178 | 1.000 | \beta_{21}=\beta_{31}=0 | | |
| 0902.0431_FO2787 | 178 | 1.000 | a_{2}=a_{3}=0 | | |
| 0902.0431_FO2788 | 178 | 1.000 | \left[\alpha \Phi, 1^{-}\right]=\left[\alpha \Phi, \alpha 1^{-}\right]= | | |
| 0902.0431_FO2789 | 178 | 1.000 | \alpha\left[\Phi, 1^{-}\right]=0 | | |
| 0902.0431_FO2790 | 178 | 1.000 | a_{1}=0 | | |
| 0902.0431_FO2791 | 178 | 1.000 | \left[\alpha P^{-}, 1\right]=\left[\alpha P^{-}, \alpha 1\right]=\alpha\left[P^{-}, 1\right]=-\alpha P^{-} | | |
| 0902.0431_FO2792 | 178 | 1.000 | \beta_{12}=\beta_{32}=0 | | |
| 0902.0431_FO2793 | 178 | 1.000 | b_{1}=b_{2}=b_{3}=0 | | |
| 0902.0431_FO2794 | 178 | 1.000 | \left[\alpha Q_{-}, 1\right]= | | |
| 0902.0431_FO2795 | 178 | 1.000 | \left[\alpha Q_{-}, \alpha 1\right]=\alpha\left[Q_{-}, 1\right]=\alpha Q_{-} | | |
| 0902.0431_FO2796 | 178 | 1.000 | \beta_{13}=\beta_{23}=0 | | |
| 0902.0431_FO2797 | 178 | 1.000 | c_{1}=c_{2}=c_{3}=0 | | |
| 0902.0431_FO2798 | 179 | 1.000 | [(0, P, 0,0,0,0),(0,0, Q, 0,0,0)]=\left(P \times Q, 0,0,-\frac{1}{8}\{P, Q\}, 0,0\right) | | |
| 0902.0431_FO2799 | 179 | 1.000 | \left[\left(0, \beta_{2} P, 0,0,0,0\right),\left(0,0, \beta_{3} Q, 0,0,0\right)\right]=\left(\beta_{1}(P \times Q), 0,0,-\frac{1}{8}\{P, Q\}, 0,0\right) | | |
| 0902.0431_FO2800 | 179 | 1.000 | [(0, P, 0,0,0,0),(0, Q, 0,0,0,0)]=\left(0,0,0, \frac{1}{4}\{P, Q\}, 0,0\right) | | |
| 0902.0431_FO2801 | 179 | 1.000 | [(\Phi, 0,0,0,0,0),(0, P, 0,0,0,0)]=(0, \Phi P, 0,0,0,0) | | |
| 0902.0431_FO2802 | 179 | 1.000 | \beta_{2}=\beta_{3} | | |
| 0902.0431_FO2803 | 179 | 1.000 | \beta^{-1} P | | |
| 0902.0431_FO2804 | 179 | 1.000 | v: \mathfrak{e}_{8}{ }^{C} \rightarrow \mathfrak{e}_{8}{ }^{C} | | |
| 0902.0431_FO2805 | 179 | 0.992 | v \in E_{8} | | |
| 0902.0431_FO2806 | 179 | 0.992 | v^{2}=1 | | |
| 0902.0431_FO2807 | 179 | 1.000 | \left(E_{8}\right)^{v} | | |
| 0902.0431_FO2808 | 180 | 1.000 | \alpha \in E_{8} | | |
| 0902.0431_FO2809 | 180 | 1.000 | \alpha 1^{-}=1^{-} | | |
| 0902.0431_FO2810 | 180 | 0.685 | \alpha 1=(\Phi, P, Q, r, s, t) | | |
| 0902.0431_FO2811 | 180 | 0.685 | \left[\alpha 1_{1} 1_{-}\right]=\left[\alpha 1, \alpha 1_{-}\right]= | | |
| 0902.0431_FO2812 | 180 | 1.000 | \alpha\left[1,1_{-}\right]=-2 \alpha 1_{-}=-21_{-} | | |
| 0902.0431_FO2813 | 180 | 1.000 | P=0, s=0, r=1 | | |
| 0902.0431_FO2814 | 180 | 1.000 | \langle\alpha 1, \alpha 1\rangle=\langle 1,1\rangle=8 | | |
| 0902.0431_FO2815 | 180 | 1.000 | \langle\Phi, \Phi\rangle+\langle Q, Q\rangle+8+4(\tau t) t=8 | | |
| 0902.0431_FO2816 | 180 | 1.000 | \Phi=0, Q=0, t=0 | | |
| 0902.0431_FO2817 | 180 | 1.000 | \left[1^{-}, 1_{-}\right]=1 | | |
| 0902.0431_FO2818 | 180 | 0.295 | \left(E_{8}\right)_{1 \_} | | |
| 0902.0431_FO2819 | 180 | 0.848 | \quad\left(E_{8}\right)_{1-} \cong E_{7} | | |
| 0902.0431_FO2820 | 180 | 0.967 | \left(E_{8}\right)_{1_{-}}=\left(E_{8}\right)_{1,1^{-}, 1_{-}} | | |
| 0902.0431_FO2821 | 180 | 1.000 | \mathfrak{W}_{1} | | |
| 0902.0431_FO2822 | 180 | 0.963 | S U(2)=\left\{A \in M(2, C) \mid\left(\tau^{t} A\right) A=E, \operatorname{det} A=1\right\} | | |
| 0902.0431_FO2823 | 180 | 1.000 | A=\left(\begin{array}{cc}a & -\tau b \\ b & \tau a\end{array}\right) \in S U(2), \varphi_{3}(A): \mathfrak{e}_{8}{ }^{C} \rightarrow \mathfrak{e}_{8}{ }^{C} | | |
| 0902.0431_FO2824 | 181 | 1.000 | A=\left(\begin{array}{cc}a & -\tau b \\ b & \tau a\end{array}\right)=\exp \left(\begin{array}{cc}-i \nu & -\tau \rho \\ \rho & i \nu\end{array}\right) \in S U(2) | | |
| 0902.0431_FO2825 | 181 | 1.000 | \phi(A)= | | |
| 0902.0431_FO2826 | 181 | 0.998 | \exp (\Theta(0,0,0, i \nu, \rho,-\tau \rho)) \in\left(E_{8}\right)^{v} | | |
| 0902.0431_FO2827 | 181 | 0.995 | \left(E_{8}\right)^{v} \cong\left(S U(2) \times E_{7}\right) / \boldsymbol{Z}_{2}, \boldsymbol{Z}_{2}=\{(E, 1),(-E,-1)\} | | |
| 0902.0431_FO2828 | 181 | 0.935 | \varphi: S U(2) \times E_{7} \rightarrow\left(E_{8}\right)^{v} | | |
| 0902.0431_FO2829 | 181 | 1.000 | \varphi_{3}(A) \in \varphi_{3}(S U(A)) | | |
| 0902.0431_FO2830 | 181 | 1.000 | \beta \in E_{7} | | |
| 0902.0431_FO2831 | 181 | 1.000 | \varphi_{*}: \mathfrak{s u}(8) \oplus \mathfrak{e}_{7} \rightarrow\left(\mathfrak{e}_{8}\right)^{v} | | |
| 0902.0431_FO2832 | 181 | 1.000 | \operatorname{Ker} \varphi=\{(E, 1),(-E,-1)\}=\boldsymbol{Z}_{2} | | |
| 0902.0431_FO2833 | 181 | 0.999 | \left(S U(2) \times E_{7}\right) / \boldsymbol{Z}_{2} \cong\left(E_{8}\right)^{v} | | |
| 0902.0431_FO2834 | 181 | 0.990 | \widetilde{\lambda} \gamma: \mathfrak{e}_{8}{ }^{C} \rightarrow \mathfrak{e}_{8}{ }^{C} | | |
| 0902.0431_FO2835 | 181 | 0.730 | \widetilde{\lambda} \gamma \in E_{8} | | |
| 0902.0431_FO2836 | 181 | 0.730 | (\widetilde{\lambda} \gamma)^{2}=1 | | |
| 0902.0431_FO2837 | 181 | 0.999 | \left(E_{8}\right)^{\widetilde{\lambda} \gamma} | | |
| 0902.0431_FO2838 | 181 | 1.000 | l: M(8, C) \rightarrow M(16, \boldsymbol{R}) | | |
| 0902.0431_FO2839 | 182 | 1.000 | I, J \in M(16, \boldsymbol{R}) | | |
| 0902.0431_FO2840 | 182 | 1.000 | I J=-J I | | |
| 0902.0431_FO2841 | 182 | 1.000 | X, Y \in M(8, C) | | |
| 0902.0431_FO2842 | 182 | 1.000 | l(X Y)=l(X) l(Y) | | |
| 0902.0431_FO2843 | 182 | 0.965 | I l(X)=l(\tau X) I, \quad J l(X)=l(X) J | | |
| 0902.0431_FO2844 | 182 | 1.000 | { }^{t} l(X)=l\left(\tau^{t} X\right) | | |
| 0902.0431_FO2845 | 182 | 0.989 | l(\mathfrak{u}(8))=\{B \in \mathfrak{s o}(16) \mid J B=B J\} | | |
| 0902.0431_FO2846 | 182 | 1.000 | B \in \mathfrak{s o}(16) | | |
| 0902.0431_FO2847 | 182 | 0.991 | D \in \mathfrak{u}(8) | | |
| 0902.0431_FO2848 | 182 | 0.991 | J l(D)=l(D) J | | |
| 0902.0431_FO2849 | 182 | 0.991 | { }^{t} l(D)=l\left(\tau^{t} D\right)=-l(D) | | |
| 0902.0431_FO2850 | 182 | 1.000 | J B=B J | | |
| 0902.0431_FO2851 | 182 | 1.000 | B=l(D), D \in M(8, C) | | |
| 0902.0431_FO2852 | 182 | 1.000 | \tau^{t} D=-D | | |
| 0902.0431_FO2853 | 182 | 1.000 | S \in \mathfrak{S}(8, C) | | |
| 0902.0431_FO2854 | 182 | 1.000 | J B=-B J | | |
| 0902.0431_FO2855 | 182 | 1.000 | B I | | |
| 0902.0431_FO2856 | 182 | 1.000 | J B I=B I J | | |
| 0902.0431_FO2857 | 182 | 1.000 | B I=l(S), S \in M(8, C) | | |
| 0902.0431_FO2858 | 182 | 1.000 | -S={ }^{t} S | | |
| 0902.0431_FO2859 | 182 | 1.000 | B=\frac{B-J B J}{2}+\frac{B+J B J}{2} | | |
| 0902.0431_FO2860 | 182 | 1.000 | \chi:\left(\mathfrak{P}^{C}\right)_{\tau \gamma} \rightarrow | | |
| 0902.0431_FO2861 | 182 | 0.950 | \mathfrak{S}(8, C) | | |
| 0902.0431_FO2862 | 182 | 0.950 | \varphi_{*}: \mathfrak{s u}(8) \rightarrow\left(\mathfrak{e}_{6}\right)^{\lambda \gamma} | | |
| 0902.0431_FO2863 | 182 | 0.999 | \varphi_{*}: \mathfrak{s p}(4) \rightarrow\left(\mathfrak{e}_{6}\right)^{\lambda \gamma} | | |
| 0902.0431_FO2864 | 182 | 0.999 | \left(\varphi_{*} D\right) X= | | |
| 0902.0431_FO2865 | 183 | 1.000 | \left[g X_{1}, g X_{2}\right] \in \mathfrak{s p}(4) | | |
| 0902.0431_FO2866 | 183 | 1.000 | S, S_{1}, S_{2} \in \mathfrak{S}(8, C) | | |
| 0902.0431_FO2867 | 183 | 0.985 | \lambda \gamma \chi^{-1}(S)=-\chi^{-1}(i S) | | |
| 0902.0431_FO2868 | 183 | 0.984 | \quad \operatorname{tr}\left(S_{1} \tau S_{2}-S_{2} \tau S_{1}\right)=4 i\left\{\chi^{-1} S_{1}, \chi^{-1} S_{2}\right\} | | |
| 0902.0431_FO2869 | 183 | 1.000 | \chi^{-1} S=P=(X, Y, \xi, \eta) | | |
| 0902.0431_FO2870 | 183 | 1.000 | \chi^{-1} S_{i}=P_{i}=\left(X_{i}, T_{i}, \xi_{i}, \eta_{i}\right), i=1,2 | | |
| 0902.0431_FO2871 | 184 | 0.999 | D=-\left[g X_{1}, g X_{2}\right]-\left[g\left(\gamma Y_{1}\right), g\left(\gamma Y_{2}\right)\right] \in \mathfrak{s p}(4) | | |
| 0902.0431_FO2872 | 185 | 0.986 | \left(\mathfrak{e}_{8}\right)^{\widetilde{\lambda} \gamma} | | |
| 0902.0431_FO2873 | 185 | 0.989 | \mathfrak{s o}(16) | | |
| 0902.0431_FO2874 | 185 | 0.989 | \zeta: \mathfrak{s o}(16) \rightarrow\left(\mathfrak{e}_{8}\right)^{\widetilde{\lambda} \gamma} | | |
| 0902.0431_FO2875 | 185 | 0.937 | D \in \mathfrak{s u}(8), S \in \mathfrak{S}(8, C), c \in \boldsymbol{R} | | |
| 0902.0431_FO2876 | 185 | 0.937 | \varphi_{*}: \mathfrak{s u}(8) \rightarrow\left(\mathfrak{e}_{7}\right)^{\lambda \gamma}, \chi:\left(\mathfrak{P}^{C}\right)_{\text {tau } \gamma} \rightarrow | | |
| 0902.0431_FO2877 | 185 | 1.000 | \left(\varphi_{*} D\right) \lambda \gamma=\lambda \gamma\left(\varphi_{*} D\right) | | |
| 0902.0431_FO2878 | 185 | 1.000 | \left(\varphi_{*} D\right) \chi^{-1} S=\chi^{-1}\left(D S+S^{t} D\right) | | |
| 0902.0431_FO2879 | 186 | 1.000 | \zeta[l(D), l(i c E)]=\zeta[D, i c E]=\zeta 0=0 | | |
| 0902.0431_FO2880 | 186 | 0.986 | \left(E_{8}\right)^{\widetilde{\lambda} \gamma} \cong S s(16) | | |
| 0902.0431_FO2881 | 187 | 1.000 | \mathfrak{e}_{8(8)} | | |
| 0902.0431_FO2882 | 187 | 0.820 | \left(\mathfrak{e}_{8(8)}\right)^{\widetilde{\lambda} \gamma} \cong S s(16) | | |
| 0902.0431_FO2883 | 187 | 0.318 | \left[\left(\mathfrak{e}_{8(8)}\right)^{\tilde{\lambda} \gamma},\left(\mathfrak{e}_{8(8)}\right)_{-\tilde{\lambda} \gamma}\right] \subset\left(\mathfrak{e}_{8(8)}\right)_{-\widetilde{\lambda} \gamma} | | |
| 0902.0431_FO2884 | 187 | 0.318 | \left(\mathfrak{e}_{8(8)}\right)^{\tilde{\lambda} \gamma} | | |
| 0902.0431_FO2885 | 187 | 0.777 | \left(\mathfrak{e}_{8(8)}\right)_{-} \widetilde{\lambda}_{\gamma} | | |
| 0902.0431_FO2886 | 187 | 0.989 | \varphi^{C} | | |
| 0902.0431_FO2887 | 187 | 0.989 | \left(\left(\left(\mathfrak{e}_{8(8)}\right)_{-\widetilde{\lambda} \gamma}\right)^{C}=\left(e_{8}{ }^{C}\right)_{-\widetilde{\lambda} \gamma}\right. | | |
| 0902.0431_FO2888 | 187 | 0.770 | \left(\mathfrak{e}_{8(8)}\right)_{-\widetilde{\lambda} \gamma} | | |
| 0902.0431_FO2889 | 187 | 0.978 | z\left(\left(E_{8}\right)^{\widetilde{\lambda} \gamma}\right) | | |
| 0902.0431_FO2890 | 187 | 0.989 | \{1, \widetilde{\lambda} \gamma\} \subset z\left(\left(E_{8}\right)^{\widetilde{\lambda} \gamma}\right) | | |
| 0902.0431_FO2891 | 187 | 0.989 | \alpha \in z\left(\left(E_{8}\right)^{\widetilde{\lambda} \gamma}\right) | | |
| 0902.0431_FO2892 | 187 | 0.711 | \left(E_{8}\right)^{\tilde{\lambda} \gamma} | | |
| 0902.0431_FO2893 | 187 | 0.711 | \left(\mathfrak{e}_{8}{ }^{C}\right)_{-\widetilde{\lambda} \gamma} | | |
| 0902.0431_FO2894 | 187 | 0.777 | B_{8}\left(R, R^{\prime}\right) | | |
| 0902.0431_FO2895 | 187 | 0.777 | \alpha: B_{8}\left(\alpha R, \alpha R^{\prime}\right)=B_{8}\left(R, R^{\prime}\right) | | |
| 0902.0431_FO2896 | 188 | 0.999 | k^{2}=1 | | |
| 0902.0431_FO2897 | 188 | 0.999 | \left(\mathfrak{e}_{8}{ }^{C}\right)^{\widetilde{\lambda} \gamma} \cong \mathfrak{s o}(16, C) | | |
| 0902.0431_FO2898 | 188 | 0.995 | \left(\mathfrak{e}_{8}{ }^{C}\right)^{\widetilde{\lambda} \gamma} | | |
| 0902.0431_FO2899 | 188 | 0.995 | \left(\mathfrak{e}_{8}{ }^{C}\right)^{-\widetilde{\lambda} \gamma} | | |
| 0902.0431_FO2900 | 188 | 0.995 | k^{2} 1=1 | | |
| 0902.0431_FO2901 | 188 | 0.998 | k=1 | | |
| 0902.0431_FO2902 | 188 | 0.998 | k=-1 | | |
| 0902.0431_FO2903 | 188 | 0.998 | \alpha=\widetilde{\lambda} \gamma | | |
| 0902.0431_FO2904 | 188 | 1.000 | \omega_{7} | | |
| 0902.0431_FO2905 | 188 | 1.000 | \omega_{8} | | |
| 0902.0431_FO2906 | 188 | 1.000 | \omega_{1}, \omega_{2}, \cdots, \omega_{8} | | |
| 0902.0431_FO2907 | 188 | 0.986 | \operatorname{Spin}(16) | | |
| 0902.0431_FO2908 | 188 | 1.000 | \Delta_{16}{ }^{+} | | |
| 0902.0431_FO2909 | 188 | 1.000 | \Delta_{16}{ }^{-} | | |
| 0902.0431_FO2910 | 188 | 0.982 | S O(16) | | |
| 0902.0431_FO2911 | 188 | 0.934 | \quad\left(E_{8}\right)^{\widetilde{\lambda} \gamma} \cong S s(16) | | |
| 0902.0431_FO2912 | 188 | 1.000 | \sigma \in F_{4} \subset E_{6} \subset E_{7} \subset E_{8} | | |
| 0902.0431_FO2913 | 188 | 1.000 | \left(E_{8}\right)^{\sigma} | | |
| 0902.0431_FO2914 | 188 | 0.996 | \left(\mathfrak{e}_{8}\right)^{\sigma} | | |
| 0902.0431_FO2915 | 188 | 0.959 | \mathfrak{s o}(16)=\left\{\left.X \in M(16, \boldsymbol{R})\right|^{t} X=-X\right\} | | |
| 0902.0431_FO2916 | 188 | 0.785 | \left(E_{8}\right)^{\widetilde{\lambda} \gamma} \cong S s(16) \cong\left(E_{8}\right)^{\sigma} | | |
| 0902.0431_FO2917 | 189 | 1.000 | \alpha \in z\left(E_{8}\right) | | |
| 0902.0431_FO2918 | 189 | 1.000 | v \alpha=\alpha v | | |
| 0902.0431_FO2919 | 189 | 1.000 | \alpha \in \varphi\left(S U(2) \times E_{7}\right) \cong | | |
| 0902.0431_FO2920 | 189 | 0.834 | \alpha \in z\left(\varphi\left(S U(2) \times E_{7}\right)\right) | | |
| 0902.0431_FO2921 | 189 | 1.000 | v \notin z\left(E_{8}\right) | | |
| 0902.0431_FO2922 | 189 | 1.000 | \left(S U(3) \times E_{6}\right) / \boldsymbol{Z}_{3} | | |
| 0902.0431_FO2923 | 189 | 1.000 | \mathfrak{e}_{8}{ }^{C},\left\langle R_{1}, R_{2}\right\rangle, \tau \widetilde{\lambda}, w | | |
| 0902.0431_FO2924 | 189 | 1.000 | 27 \times 3=78 | | |
| 0902.0431_FO2925 | 189 | 1.000 | \left(\mathfrak{J}^{C}\right)^{3} | | |
| 0902.0431_FO2926 | 189 | 1.000 | (\boldsymbol{X}, \boldsymbol{Y}) | | |
| 0902.0431_FO2927 | 189 | 1.000 | \langle\boldsymbol{X}, \boldsymbol{Y}\rangle | | |
| 0902.0431_FO2928 | 189 | 0.996 | \boldsymbol{X} \times \boldsymbol{Y} | | |
| 0902.0431_FO2929 | 189 | 0.996 | \boldsymbol{X} \cdot \boldsymbol{Y} | | |
| 0902.0431_FO2930 | 189 | 0.996 | \boldsymbol{X} \vee \boldsymbol{Y} | | |
| 0902.0431_FO2931 | 189 | 0.972 | \boldsymbol{X}=\left(\begin{array}{l}X_{1} \\ X_{2} \\ X_{3}\end{array}\right), \boldsymbol{Y}=\left(\begin{array}{l}Y_{1} \\ Y_{2} \\ Y_{3}\end{array}\right) \in\left(\mathfrak{J}^{C}\right)^{3} | | |
| 0902.0431_FO2932 | 189 | 0.972 | \phi \in \operatorname{Hom}_{C}\left(\mathfrak{J}^{C}\right), D=\left(d_{i j}\right) \in | | |
| 0902.0431_FO2933 | 189 | 0.983 | M(3, C) | | |
| 0902.0431_FO2934 | 189 | 0.983 | \boldsymbol{X}=\left(\begin{array}{c}X_{1} \\ X_{2} \\ X_{3}\end{array}\right) \in\left(\mathfrak{J}^{C}\right)^{3} | | |
| 0902.0431_FO2935 | 189 | 0.983 | \phi \boldsymbol{X}, D \boldsymbol{X} \in\left(\mathfrak{J}^{C}\right)^{3} | | |
| 0902.0431_FO2936 | 190 | 0.999 | 8+78+27 \times 3+27 \times 3=248 | | |
| 0902.0431_FO2937 | 190 | 0.902 | \widetilde{\mathfrak{e}}_{8}^{C}=\mathfrak{e}_{7}^{C} \oplus \mathfrak{P}^{C} \oplus \mathfrak{P}^{C} \oplus C \oplus C \oplus C | | |
| 0902.0431_FO2938 | 190 | 0.900 | f: \widetilde{\mathfrak{e}}_{8}^{C} \rightarrow \mathfrak{e}_{8}^{C} | | |
| 0902.0431_FO2939 | 190 | 0.699 | \widetilde{\mathfrak{e}}_{8}{ }^{C} \cong \mathfrak{e}_{8}{ }^{C} | | |
| 0902.0431_FO2940 | 190 | 0.999 | \widetilde{\mathfrak{e}}_{8}^{C} | | |
| 0902.0431_FO2941 | 190 | 0.655 | \left(R_{i}=\left(D_{i}, \phi_{i}, \boldsymbol{X}_{i}, \boldsymbol{Y}_{i}\right) \in \mathfrak{e}_{8}{ }^{C}\right) | | |
| 0902.0431_FO2942 | 190 | 1.000 | \tau \widetilde{\lambda} | | |
| 0902.0431_FO2943 | 191 | 1.000 | w \in E_{8} | | |
| 0902.0431_FO2944 | 191 | 1.000 | \left(E_{8}\right)^{w} | | |
| 0902.0431_FO2945 | 191 | 0.545 | \left(E_{8}\right)^{w} \cong\left(S U(3) \times E_{6}\right) / \boldsymbol{Z}_{3}, \quad \boldsymbol{Z}_{3}=\left\{(E, 1),\left(\omega E, \omega^{2} 1\right),\left(\omega^{2} E\right.\right. | | |
| 0902.0431_FO2946 | 191 | 0.991 | \omega 1)\} | | |
| 0902.0431_FO2947 | 191 | 1.000 | \varphi_{1}: S U(3) \rightarrow\left(E_{8}\right)^{w} | | |
| 0902.0431_FO2948 | 191 | 0.990 | \varphi_{1}(A) \in\left(E_{8}\right)^{w} | | |
| 0902.0431_FO2949 | 191 | 0.990 | D_{1}=\left(D_{1}, 0,0,0\right) \in | | |
| 0902.0431_FO2950 | 191 | 0.994 | \mathfrak{s} \mathfrak{u}(3) \subset \mathfrak{s l}(3, C) \subset \mathfrak{e}_{8}{ }^{C} | | |
| 0902.0431_FO2951 | 191 | 0.985 | A=\exp D_{1} | | |
| 0902.0431_FO2952 | 191 | 0.985 | \varphi_{1}(A)=\exp \left(\operatorname{ad}\left(D_{1}\right)\right) \in \operatorname{Aut}\left(\mathfrak{e}_{8}{ }^{C}\right) | | |
| 0902.0431_FO2953 | 191 | 1.000 | \varphi_{1}(A) \in E_{8} | | |
| 0902.0431_FO2954 | 191 | 1.000 | w \varphi_{1}(A)=\varphi_{1}(A) w | | |
| 0902.0431_FO2955 | 191 | 1.000 | \varphi_{2}: E_{6} \rightarrow\left(E_{8}\right)^{w} | | |
| 0902.0431_FO2956 | 191 | 1.000 | \varphi_{2}(\alpha) \in\left(E_{8}\right)^{w} | | |
| 0902.0431_FO2957 | 191 | 1.000 | \phi^{\prime}= | | |
| 0902.0431_FO2958 | 191 | 1.000 | \left(0, \phi^{\prime}, 0,0\right) \in \mathfrak{e}_{6} \subset \mathfrak{e}_{6}{ }^{C} \subset \mathfrak{e}_{8}{ }^{C} | | |
| 0902.0431_FO2959 | 191 | 0.990 | \alpha=\exp \phi^{\prime} | | |
| 0902.0431_FO2960 | 191 | 0.990 | \varphi_{2}(\alpha)=\exp \left(\operatorname{ad}\left(\phi^{\prime}\right)\right) \in \operatorname{Aut}\left(\mathfrak{e}_{8}{ }^{C}\right) | | |
| 0902.0431_FO2961 | 192 | 1.000 | \varphi_{2}(\alpha) \in E_{8} | | |
| 0902.0431_FO2962 | 192 | 1.000 | w \varphi_{2}(\alpha)=\varphi_{2}(\alpha) w | | |
| 0902.0431_FO2963 | 192 | 0.798 | \varphi: S U(3) \times E_{6} \rightarrow\left(E_{8}\right)^{w} | | |
| 0902.0431_FO2964 | 192 | 1.000 | \varphi_{1}(A) | | |
| 0902.0431_FO2965 | 192 | 1.000 | \varphi_{2}(\alpha) | | |
| 0902.0431_FO2966 | 192 | 0.999 | \operatorname{Ker} \varphi=\left\{(E, 1),\left(\omega E, \omega^{2} 1\right),\left(\omega^{2} E, \omega 1\right)\right\}=\boldsymbol{Z}_{3} | | |
| 0902.0431_FO2967 | 192 | 0.996 | \left(S U(3) \times E_{6}\right) / \boldsymbol{Z}_{3} \cong\left(E_{8}\right)^{w} | | |
| 0902.0431_FO2968 | 192 | 1.000 | \Lambda^{k}\left(C^{n}\right) | | |
| 0902.0431_FO2969 | 192 | 1.000 | \boldsymbol{e}_{1}, \cdots, \boldsymbol{e}_{n} | | |
| 0902.0431_FO2970 | 192 | 0.755 | n | | |
| 0902.0431_FO2971 | 192 | 0.755 | C^{n} | | |
| 0902.0431_FO2972 | 192 | 0.755 | (\boldsymbol{x}, \boldsymbol{y}) | | |
| 0902.0431_FO2973 | 192 | 1.000 | \left(\boldsymbol{e}_{i}, \boldsymbol{e}_{j}\right)=\delta_{i j} | | |
| 0902.0431_FO2974 | 192 | 0.985 | \boldsymbol{e}_{i_{1}} \wedge \cdots \wedge \boldsymbol{e}_{i_{k}}, i_{1}<\cdots<i_{k} | | |
| 0902.0431_FO2975 | 192 | 1.000 | \boldsymbol{u} \in \Lambda^{k}\left(C^{n}\right) | | |
| 0902.0431_FO2976 | 192 | 1.000 | * \boldsymbol{u} \in \Lambda^{n-k}\left(C^{n}\right) | | |
| 0902.0431_FO2977 | 192 | 1.000 | S L(n, C) | | |
| 0902.0431_FO2978 | 193 | 1.000 | \mathfrak{s l}(n, C) | | |
| 0902.0431_FO2979 | 193 | 0.997 | A \in S L(n, C), D \in \mathfrak{s l}(n, C) | | |
| 0902.0431_FO2980 | 193 | 0.997 | \boldsymbol{u}, \boldsymbol{v} \in \Lambda^{k}\left(C^{n}\right) | | |
| 0902.0431_FO2981 | 193 | 0.643 | \left(A \boldsymbol{u},{ }^{t} A^{-1} \boldsymbol{v}\right)=(\boldsymbol{u}, \boldsymbol{v}), \quad(D \boldsymbol{u}, \boldsymbol{v})+\left(\boldsymbol{u},-{ }^{t} D \boldsymbol{v}\right)=0 | | |
| 0902.0431_FO2982 | 193 | 0.739 | *(A \boldsymbol{u})={ }^{t} A^{-1}(* \boldsymbol{u}), \quad *(D \boldsymbol{u})=-{ }^{t} D^{-1}(* \boldsymbol{u}) | | |
| 0902.0431_FO2983 | 193 | 1.000 | \boldsymbol{u}, \boldsymbol{v} \in \Lambda^{k}\left(C^{n}\right)(1 \leq k \leq n) | | |
| 0902.0431_FO2984 | 193 | 1.000 | \boldsymbol{u} \times \boldsymbol{v} | | |
| 0902.0431_FO2985 | 193 | 0.950 | \operatorname{tr}(\boldsymbol{u} \times \boldsymbol{v})=0, \boldsymbol{u} \times \boldsymbol{v} | | |
| 0902.0431_FO2986 | 193 | 0.950 | \mathfrak{s} \mathfrak{l}(n, C) | | |
| 0902.0431_FO2987 | 193 | 0.732 | A(\boldsymbol{u} \times \boldsymbol{v}) A^{-1}=A \boldsymbol{u} \times{ }^{t} A^{-1} \boldsymbol{v},[D, \boldsymbol{u} \times \boldsymbol{v}]=D \boldsymbol{u} \times \boldsymbol{v}+\boldsymbol{u} \times\left(-{ }^{t} D \boldsymbol{v}\right) | | |
| 0902.0431_FO2988 | 193 | 1.000 | { }^{t}(\boldsymbol{u} \times \boldsymbol{v})=\boldsymbol{v} \times \boldsymbol{u}, \quad \tau(\boldsymbol{u} \times \boldsymbol{v})=\tau(\boldsymbol{u}) \times \tau(\boldsymbol{v}) | | |
| 0902.0431_FO2989 | 193 | 1.000 | \operatorname{tr}(D(\boldsymbol{u} \times \boldsymbol{v}))=(-1)^{n-k}(D \boldsymbol{u}, \boldsymbol{v}) | | |
| 0902.0431_FO2990 | 193 | 1.000 | 80+84+84=248 | | |
| 0902.0431_FO2991 | 193 | 1.000 | \boldsymbol{u}, \boldsymbol{v}, \boldsymbol{w} \in \Lambda^{3}\left(C^{9}\right) | | |
| 0902.0431_FO2992 | 193 | 0.926 | \quad \boldsymbol{u} \times *(\boldsymbol{v} \wedge \boldsymbol{w})+\boldsymbol{v} \times *(\boldsymbol{w} \wedge \boldsymbol{u})+\boldsymbol{w} \times *(\boldsymbol{u} \wedge \boldsymbol{v})=0 | | |
| 0902.0431_FO2993 | 193 | 0.800 | (\boldsymbol{u} \times \boldsymbol{w}) \boldsymbol{v}-(\boldsymbol{v} \times \boldsymbol{w}) \boldsymbol{u}+*(*(\boldsymbol{u} \times \boldsymbol{v}) \wedge \boldsymbol{w})=0 | | |
| 0902.0431_FO2994 | 194 | 0.999 | \boldsymbol{u}=\boldsymbol{u}_{1} \wedge \boldsymbol{u}_{2} \wedge \boldsymbol{u}_{3}, \boldsymbol{v}=\boldsymbol{u}_{4} \wedge \boldsymbol{u}_{5} \wedge \boldsymbol{u}_{6} | | |
| 0902.0431_FO2995 | 194 | 0.999 | \boldsymbol{w}=\boldsymbol{u}_{7} \wedge \boldsymbol{u}_{8} \wedge \boldsymbol{u}_{9} | | |
| 0902.0431_FO2996 | 194 | 0.998 | \boldsymbol{x}, \boldsymbol{y} \in C^{9} | | |
| 0902.0431_FO2997 | 194 | 1.000 | \boldsymbol{x}=\sum_{i=1}^{9} x_{i} \boldsymbol{e}_{i}, \boldsymbol{u}_{j}=\sum_{k=1}^{9} u_{j k} \boldsymbol{e}_{k} | | |
| 0902.0431_FO2998 | 194 | 1.000 | U=\left(u_{j k}\right) \in M(9, C) | | |
| 0902.0431_FO2999 | 194 | 1.000 | \widetilde{u}_{j k} | | |
| 0902.0431_FO3000 | 194 | 1.000 | u_{j k} | | |
| 0902.0431_FO3001 | 194 | 1.000 | U | | |
| 0902.0431_FO3002 | 194 | 1.000 | \boldsymbol{u}=\boldsymbol{u}_{1} \wedge \boldsymbol{u}_{2} \wedge \boldsymbol{u}_{3} | | |
| 0902.0431_FO3003 | 194 | 1.000 | \boldsymbol{v}=\boldsymbol{v}_{1} \wedge \boldsymbol{v}_{2} \wedge \boldsymbol{v}_{3} | | |
| 0902.0431_FO3004 | 194 | 1.000 | \boldsymbol{a} \in \Lambda^{3}\left(C^{9}\right) | | |
| 0902.0431_FO3005 | 195 | 0.997 | \mathfrak{e}_{8}{ }^{C}=\mathfrak{s l}(9, C) \oplus \Lambda^{3}\left(C^{9}\right) \oplus \Lambda^{3}\left(C^{9}\right) | | |
| 0902.0431_FO3006 | 195 | 1.000 | I=\{i, j, k\}(i<j<k) | | |
| 0902.0431_FO3007 | 195 | 1.000 | \{1,2, \cdots, 9\} | | |
| 0902.0431_FO3008 | 195 | 1.000 | \mathfrak{g}=\mathfrak{e}_{8}{ }^{C} | | |
| 0902.0431_FO3009 | 195 | 1.000 | \mathfrak{s l}(9, C) \cap \mathfrak{a}=\{0\} | | |
| 0902.0431_FO3010 | 195 | 1.000 | \mathfrak{q} \cap \mathfrak{a}=\{0\} | | |
| 0902.0431_FO3011 | 195 | 1.000 | p: \mathfrak{g} \rightarrow \mathfrak{s l}(9, C) | | |
| 0902.0431_FO3012 | 195 | 1.000 | p(\mathfrak{a})=0 | | |
| 0902.0431_FO3013 | 195 | 1.000 | \mathfrak{q} | | |
| 0902.0431_FO3014 | 195 | 1.000 | \mathfrak{s l}(9, C) | | |
| 0902.0431_FO3015 | 195 | 1.000 | p(\mathfrak{a})=\mathfrak{s l}(9, C) | | |
| 0902.0431_FO3016 | 195 | 1.000 | D=\sum_{i=1}^{8} H_{i} \in | | |
| 0902.0431_FO3017 | 195 | 0.831 | \mathfrak{s} \mathfrak{l}(9, C), H_{i}=E_{i i}-E_{99} | | |
| 0902.0431_FO3018 | 195 | 0.831 | (\boldsymbol{u}, \boldsymbol{v})=\left(\sum_{I} u_{I} \boldsymbol{e}_{I}, \sum_{J} v_{J} \boldsymbol{e}_{J}\right) \in \mathfrak{q} | | |
| 0902.0431_FO3019 | 195 | 0.998 | (D, \boldsymbol{u}, \boldsymbol{v}) \in \mathfrak{a} | | |
| 0902.0431_FO3020 | 195 | 0.998 | [(D, 0,0),(X, \boldsymbol{u}, \boldsymbol{v})]=\left(0, D \boldsymbol{u},-{ }^{t} D \boldsymbol{v}\right) \in \mathfrak{q} \cap \mathfrak{a}=\{0\} | | |
| 0902.0431_FO3021 | 195 | 0.882 | u_{I}=0 | | |
| 0902.0431_FO3022 | 195 | 0.882 | v_{J}=0 | | |
| 0902.0431_FO3023 | 195 | 0.882 | 0 \neq(D, \boldsymbol{u}, \boldsymbol{v})=(D, 0,0) \in \mathfrak{s} \mathfrak{l}(9, C) \cap \mathfrak{a}=\{0\} | | |
| 0902.0431_FO3024 | 195 | 1.000 | \mathfrak{s l}(9, C) \cap \mathfrak{a} \neq\{0\} | | |
| 0902.0431_FO3025 | 195 | 1.000 | \mathfrak{s l}(9, C) \cap \mathfrak{a} | | |
| 0902.0431_FO3026 | 195 | 0.999 | \mathfrak{s l}(9, C) \subset \mathfrak{a} | | |
| 0902.0431_FO3027 | 195 | 0.999 | \boldsymbol{e}_{i} \wedge \boldsymbol{e}_{j} \wedge \boldsymbol{e}_{k} \in \Lambda^{3}\left(C^{9}\right) | | |
| 0902.0431_FO3028 | 195 | 1.000 | (D, 0,0) \in \mathfrak{s l}(9, C) \subset \mathfrak{a} | | |
| 0902.0431_FO3029 | 196 | 1.000 | \mathfrak{q} \subset \mathfrak{a} | | |
| 0902.0431_FO3030 | 196 | 1.000 | \mathfrak{a}=\mathfrak{g} | | |
| 0902.0431_FO3031 | 196 | 1.000 | \mathfrak{q} \cap \mathfrak{a} \neq\{0\} | | |
| 0902.0431_FO3032 | 196 | 1.000 | R=(0, \boldsymbol{u}, \boldsymbol{v}) | | |
| 0902.0431_FO3033 | 196 | 1.000 | \mathfrak{q} \cap \mathfrak{a} | | |
| 0902.0431_FO3034 | 196 | 1.000 | \boldsymbol{u} \neq 0 | | |
| 0902.0431_FO3035 | 196 | 1.000 | \boldsymbol{u}=\sum_{I} u_{I} \boldsymbol{e}_{I} | | |
| 0902.0431_FO3036 | 196 | 1.000 | u_{\{123\}}=1 | | |
| 0902.0431_FO3037 | 196 | 1.000 | S_{i j}=\left(E_{i i}-E_{j j}, 0,0\right) \in \mathfrak{g} | | |
| 0902.0431_FO3038 | 196 | 1.000 | T=\left(0,0, \boldsymbol{e}_{1} \wedge \boldsymbol{e}_{2} \wedge \boldsymbol{e}_{4}\right) \in \mathfrak{g} | | |
| 0902.0431_FO3039 | 196 | 1.000 | \boldsymbol{v} \neq 0 | | |
| 0902.0431_FO3040 | 196 | 0.997 | \mathfrak{e}_{8}{ }^{C}=\mathfrak{s l}(9, C) \oplus | | |
| 0902.0431_FO3041 | 196 | 1.000 | \Lambda^{3}\left(C^{9}\right) \oplus \Lambda^{3}\left(C^{9}\right) | | |
| 0902.0431_FO3042 | 196 | 1.000 | B | | |
| 0902.0431_FO3043 | 196 | 1.000 | B_{8}\left(R_{1}, R_{2}\right)=k B\left(R_{1}, R_{2}\right) | | |
| 0902.0431_FO3044 | 196 | 1.000 | R_{i} \in \mathfrak{e}_{8}{ }^{C} | | |
| 0902.0431_FO3045 | 196 | 1.000 | R=R_{1}=R_{2}=\left(E_{11}-E_{22}, 0,0\right) \in \mathfrak{e}_{8}{ }^{C} | | |
| 0902.0431_FO3046 | 196 | 1.000 | k=60 | | |
| 0902.0431_FO3047 | 197 | 1.000 | w_{3} \in E_{8} | | |
| 0902.0431_FO3048 | 197 | 1.000 | w_{3}{ }^{3}=1 | | |
| 0902.0431_FO3049 | 197 | 1.000 | \left(E_{8}\right)^{w_{3}} | | |
| 0902.0431_FO3050 | 197 | 0.907 | \quad\left(E_{8}\right)^{w_{3}} \cong S U(9) / \boldsymbol{Z}_{3}, \quad \boldsymbol{Z}_{3}=\left\{E, \omega E, \omega^{2} E\right\} | | |
| 0902.0431_FO3051 | 197 | 1.000 | \varphi: S U(9) \rightarrow\left(E_{8}\right)^{w_{3}} | | |
| 0902.0431_FO3052 | 197 | 0.903 | \varphi(A) \in\left(E_{8}\right)^{w_{3}} | | |
| 0902.0431_FO3053 | 197 | 0.903 | A=\exp X, X \in \mathfrak{s u}(9) | | |
| 0902.0431_FO3054 | 197 | 1.000 | \varphi(A) \in E_{8} | | |
| 0902.0431_FO3055 | 197 | 1.000 | w_{3} \varphi(A)=\varphi(A) w_{3} | | |
| 0902.0431_FO3056 | 197 | 0.996 | \left(\mathfrak{e}_{8}\right)^{w_{3}} | | |
| 0902.0431_FO3057 | 197 | 0.998 | \operatorname{ker} \varphi=\left\{E, \omega E, \omega^{2} E\right\}=\boldsymbol{Z}_{3} | | |
| 0902.0431_FO3058 | 197 | 0.608 | S U(9) / \boldsymbol{Z}_{3} \cong\left(E_{8}\right)^{w_{3}} | | |
| 0902.0431_FO3059 | 197 | 1.000 | 48+50 \times 4=248 | | |
| 0902.0431_FO3060 | 198 | 1.000 | \boldsymbol{x}, \boldsymbol{y}, \boldsymbol{z} \in \Lambda^{1}\left(C^{5}\right)=C^{5} | | |
| 0902.0431_FO3061 | 198 | 1.000 | \boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c} \in \Lambda^{2}\left(C^{5}\right) | | |
| 0902.0431_FO3062 | 198 | 0.514 | \quad * \boldsymbol{a} \wedge *(\boldsymbol{b} \wedge \boldsymbol{c})+* \boldsymbol{b} \wedge *(\boldsymbol{c} \wedge \boldsymbol{a})+* \boldsymbol{c} \wedge *(\boldsymbol{a} \wedge \boldsymbol{b})=0 | | |
| 0902.0431_FO3063 | 198 | 0.928 | \quad *(\boldsymbol{a} \wedge *(* \boldsymbol{b} \wedge \boldsymbol{x}))+*(\boldsymbol{b} \wedge *(* \boldsymbol{a} \wedge \boldsymbol{x}))+\boldsymbol{x} \wedge *(\boldsymbol{a} \wedge \boldsymbol{b})=0 | | |
| 0902.0431_FO3064 | 198 | 0.740 | *(*(\boldsymbol{x} \wedge \boldsymbol{y}) \wedge \boldsymbol{z})=(\boldsymbol{x}, \boldsymbol{z}) \boldsymbol{y}-(\boldsymbol{y}, \boldsymbol{z}) \boldsymbol{x} | | |
| 0902.0431_FO3065 | 198 | 1.000 | \boldsymbol{x} \wedge *(* \boldsymbol{a} \wedge \boldsymbol{y})+*(\boldsymbol{y} \wedge+(\boldsymbol{a} \wedge \boldsymbol{x}))-(\boldsymbol{x}, \boldsymbol{y}) \boldsymbol{a}=0 | | |
| 0902.0431_FO3066 | 198 | 0.658 | \quad *(\boldsymbol{a} \wedge *(\boldsymbol{b} \wedge \boldsymbol{x}))-*(* \boldsymbol{b} \wedge *(* \boldsymbol{a} \wedge \boldsymbol{x}))-(\boldsymbol{a}, \boldsymbol{b}) \boldsymbol{x}=0 | | |
| 0902.0431_FO3067 | 198 | 0.974 | \quad \boldsymbol{a} \times *(\boldsymbol{b} \wedge \boldsymbol{x})+\boldsymbol{b} \times *(\boldsymbol{a} \wedge \boldsymbol{x})-\boldsymbol{x} \times *(\boldsymbol{a} \wedge \boldsymbol{b})=0 | | |
| 0902.0431_FO3068 | 198 | 0.748 | \quad *(* \boldsymbol{a} \wedge \boldsymbol{x}) \times \boldsymbol{y}-*(* \boldsymbol{a} \wedge \boldsymbol{y}) \times \boldsymbol{x}+\boldsymbol{a} \times(\boldsymbol{x} \wedge \boldsymbol{y})=0 | | |
| 0902.0431_FO3069 | 198 | 0.999 | (\boldsymbol{a} \wedge \boldsymbol{b}) \boldsymbol{c}=*(*(\boldsymbol{a} \wedge \boldsymbol{c}) \wedge \boldsymbol{b})-\frac{1}{5}(\boldsymbol{a}, \boldsymbol{b}) \boldsymbol{c}-(\boldsymbol{b}, \boldsymbol{c}) \boldsymbol{a} | | |
| 0902.0431_FO3070 | 198 | 0.643 | \quad(\boldsymbol{x} \times \boldsymbol{y}) \boldsymbol{a}=-*(\boldsymbol{y} \wedge *(\boldsymbol{x} \wedge \boldsymbol{a}))+\frac{3}{5}(\boldsymbol{x}, \boldsymbol{y}) \boldsymbol{a} | | |
| 0902.0431_FO3071 | 198 | 0.999 | \boldsymbol{a}=\boldsymbol{a}_{1} \wedge \boldsymbol{a}_{2}, \boldsymbol{b}=\boldsymbol{a}_{3} \wedge \boldsymbol{a}_{4}, \boldsymbol{c}=\boldsymbol{a}_{5} \wedge \boldsymbol{a}_{6} | | |
| 0902.0431_FO3072 | 198 | 0.999 | \boldsymbol{a}_{i}=\sum_{j=1}^{5} a_{i j} \boldsymbol{e}_{j} | | |
| 0902.0431_FO3073 | 199 | 1.000 | \boldsymbol{v} \in \Lambda^{1}\left(C^{5}\right)=C^{5} | | |
| 0902.0431_FO3074 | 199 | 1.000 | \boldsymbol{a}=\boldsymbol{a}_{1} \wedge \boldsymbol{a}_{2} | | |
| 0902.0431_FO3075 | 199 | 0.999 | \boldsymbol{v}, \boldsymbol{w} \in \Lambda^{1}\left(C^{5}\right)=C^{5} | | |
| 0902.0431_FO3076 | 200 | 1.000 | \boldsymbol{c}=\boldsymbol{c}_{1} \wedge \boldsymbol{c}_{2} | | |
| 0902.0431_FO3077 | 201 | 1.000 | \boldsymbol{d} \in \Lambda^{2}\left(C^{5}\right) | | |
| 0902.0431_FO3078 | 201 | 0.591 | \mathfrak{e}_{8}^{C}=\mathfrak{g}_{0} \oplus \mathfrak{g}_{1} \oplus \mathfrak{g}_{2} \oplus \mathfrak{g}_{-2} \oplus \mathfrak{g}_{-1} | | |
| 0902.0431_FO3079 | 201 | 1.000 | \mathfrak{g}=\mathfrak{e}_{8}{ }^{C}=\mathfrak{g}_{01} \oplus \mathfrak{g}_{12} \oplus \mathfrak{q} | | |
| 0902.0431_FO3080 | 201 | 0.999 | \mathfrak{g}_{01} \cap \mathfrak{a}=\{0\}, \mathfrak{g}_{02} \cap \mathfrak{a}=\{0\} | | |
| 0902.0431_FO3081 | 201 | 0.999 | p_{i}: \mathfrak{g} \rightarrow \mathfrak{g}_{0 i} | | |
| 0902.0431_FO3082 | 201 | 1.000 | (i=1,2) | | |
| 0902.0431_FO3083 | 201 | 1.000 | p_{1}(\mathfrak{a})=\{0\} | | |
| 0902.0431_FO3084 | 201 | 1.000 | p_{2}(\mathfrak{a})=\{0\} | | |
| 0902.0431_FO3085 | 201 | 1.000 | p_{1}(\mathfrak{a})=\mathfrak{g}_{01} | | |
| 0902.0431_FO3086 | 201 | 1.000 | \mathfrak{g}_{01} | | |
| 0902.0431_FO3087 | 201 | 1.000 | C=\sum_{i=1}^{4} H_{i} \in | | |
| 0902.0431_FO3088 | 201 | 0.531 | H_{i}=E_{i i}-E_{55} | | |
| 0902.0431_FO3089 | 201 | 0.531 | \left(D, g_{1}, g_{2}, g_{-2}, g_{-1}\right) \in \mathfrak{g}_{01} \oplus \mathfrak{q} | | |
| 0902.0431_FO3090 | 201 | 1.000 | \left(C, D, g_{1}, g_{2}, g_{-2}, g_{-1}\right) \in \mathfrak{a} | | |
| 0902.0431_FO3091 | 201 | 0.869 | \left[C, g_{i}\right]=0(i=1,2,-2,-1) | | |
| 0902.0431_FO3092 | 201 | 0.869 | \operatorname{ad} X | | |
| 0902.0431_FO3093 | 201 | 1.000 | g_{i}=0 | | |
| 0902.0431_FO3094 | 201 | 1.000 | (C, D) \in \mathfrak{g}_{0} \cap \mathfrak{a} | | |
| 0902.0431_FO3095 | 202 | 1.000 | \mathfrak{g}_{01} \cap \mathfrak{a} \neq\{0\} | | |
| 0902.0431_FO3096 | 202 | 1.000 | \mathfrak{g}_{02} \cap \mathfrak{a} \neq\{0\} | | |
| 0902.0431_FO3097 | 202 | 1.000 | \mathfrak{g}_{01} \subset \mathfrak{a} | | |
| 0902.0431_FO3098 | 202 | 1.000 | \left[\mathfrak{g}_{01}, \mathfrak{g}_{i}\right]=\mathfrak{g}_{i}(i=1,2,-2,-1) | | |
| 0902.0431_FO3099 | 202 | 1.000 | \mathfrak{q} \subset \boldsymbol{a} | | |
| 0902.0431_FO3100 | 202 | 1.000 | \mathfrak{g}_{02} \subset \mathfrak{a} | | |
| 0902.0431_FO3101 | 202 | 1.000 | R=\left(g_{1}, g_{2}, g_{-2}, g_{-1}\right)\left(g_{i} \in \mathfrak{g}_{i}\right) | | |
| 0902.0431_FO3102 | 202 | 1.000 | g_{1} \neq 0 | | |
| 0902.0431_FO3103 | 202 | 1.000 | g_{1}=\sum_{i, j<k} g_{i j k} \boldsymbol{e}_{i} \otimes\left(\boldsymbol{e}_{j} \wedge \boldsymbol{e}_{k}\right) | | |
| 0902.0431_FO3104 | 202 | 1.000 | g_{112} \neq 0 | | |
| 0902.0431_FO3105 | 202 | 1.000 | S_{i j k l}=\left(E_{i i}-E_{j j}, E_{k k}-E_{l l}\right) \in \mathfrak{g}_{0} | | |
| 0902.0431_FO3106 | 202 | 1.000 | T=\boldsymbol{e}_{2} \otimes \boldsymbol{e}_{1} \wedge \boldsymbol{e}_{2} \in \mathfrak{g}_{-1} | | |
| 0902.0431_FO3107 | 202 | 1.000 | g_{i} \neq 0(i=2,-2,-1) | | |
| 0902.0431_FO3108 | 202 | 0.985 | \mathfrak{e}_{8}{ }^{C}=\mathfrak{s l}(5, C) \oplus | | |
| 0902.0431_FO3109 | 202 | 0.999 | \mathfrak{s} \mathfrak{l}(5, C) \oplus \mathfrak{g}_{1} \oplus \mathfrak{g}_{2} \oplus \mathfrak{g}_{-2} \oplus \mathfrak{g}_{-1} | | |
| 0902.0431_FO3110 | 202 | 0.657 | R_{i}=\left(C_{i}, D_{i}, \boldsymbol{x}_{i} \otimes \boldsymbol{a}_{i}, \boldsymbol{b}_{i} \otimes \boldsymbol{y}_{i}, \boldsymbol{c}_{i} \otimes \boldsymbol{z}_{i}, \boldsymbol{w}_{i} \otimes \boldsymbol{d}_{i}\right) \in \mathfrak{e}_{8}^{C} | | |
| 0902.0431_FO3111 | 202 | 1.000 | R=R_{1}=R_{2}=\left(E_{11}-E_{22}, 0,0,0,0,0\right) \in \mathfrak{e}_{8}{ }^{C} | | |
| 0902.0431_FO3112 | 203 | 1.000 | \zeta=\exp (2 \pi i / 5) \in C | | |
| 0902.0431_FO3113 | 203 | 1.000 | z_{5} \in E_{8} | | |
| 0902.0431_FO3114 | 203 | 1.000 | z_{5}{ }^{5}=1 | | |
| 0902.0431_FO3115 | 203 | 1.000 | \left(E_{8}\right)^{z_{5}} | | |
| 0902.0431_FO3116 | 203 | 0.998 | \left(E_{8}\right)^{z_{5}} \cong(S U(5) \times S U(5)) / \boldsymbol{Z}_{5}, \boldsymbol{Z}_{5}=\left\{(E, E),\left(\zeta E, \zeta^{2} E\right)\right. | | |
| 0902.0431_FO3117 | 203 | 0.520 | \left.\left(\zeta^{2} E, \zeta^{4} E\right),\left(\zeta^{3} E, \zeta E\right),\left(\zeta^{4} E, \zeta^{3} E\right)\right\}, \zeta=\exp (2 \pi i / 5) | | |
| 0902.0431_FO3118 | 203 | 0.936 | \varphi_{1}, \varphi_{2}: S U(5) \rightarrow E_{8} | | |
| 0902.0431_FO3119 | 203 | 1.000 | \varphi_{1} | | |
| 0902.0431_FO3120 | 203 | 1.000 | \varphi_{2} | | |
| 0902.0431_FO3121 | 203 | 1.000 | \varphi_{1}(A), \varphi_{2}(B) \in E_{8} | | |
| 0902.0431_FO3122 | 203 | 1.000 | Z \in \mathfrak{s u}(5) | | |
| 0902.0431_FO3123 | 203 | 1.000 | (Z, 0) \in \mathfrak{g}_{0} | | |
| 0902.0431_FO3124 | 203 | 1.000 | \varphi_{1}(A) \in \operatorname{Aut}\left(\mathfrak{e}_{8}{ }^{C}\right)=E_{8}{ }^{C} | | |
| 0902.0431_FO3125 | 204 | 1.000 | \varphi_{2}(B) \in E_{8} | | |
| 0902.0431_FO3126 | 204 | 0.981 | \varphi: S U(5) \times S U(5) \rightarrow E_{8} | | |
| 0902.0431_FO3127 | 204 | 1.000 | \varphi_{2}(B) | | |
| 0902.0431_FO3128 | 204 | 1.000 | \left(\mathfrak{e}_{8}\right)^{z_{5}}=\mathfrak{s u}(5) \oplus \mathfrak{s u}(5) | | |
| 0902.0431_FO3129 | 204 | 1.000 | (S U(5) \times | | |
| 0902.0431_FO3130 | 204 | 0.546 | S U(5)) / \boldsymbol{Z}_{5} \cong\left(E_{8}\right)^{z_{5}} | | |
| 0902.0431_FO3131 | 204 | 0.971 | R_{1}, R_{2} \in \mathfrak{e}_{8(8)} | | |
| 0902.0431_FO3132 | 204 | 0.971 | \mathfrak{e}_{8(-24)} | | |
| 0902.0431_FO3133 | 205 | 0.574 | E_{6} /(U(1) \operatorname{Spin}(10)), E_{7} /\left(U(1) E_{6}\right) | | |
| 0902.0431_FO3134 | 205 | 1.000 | E_{8} /\left(U(1) E_{7}\right) | | |
| 0902.0431_FO3135 | 205 | 1.000 | E_{7(-133)} | | |
| 0902.0431_FO3136 | 205 | 1.000 | E_{8(-248)} | | |
| 0902.0431_FO3137 | 206 | 0.946 | \mathfrak{g}_{0} | | |
| 0902.0431_FO3138 | 206 | 0.946 | \mathfrak{g}_{e v} | | |
| 0902.0431_FO3139 | 206 | 1.000 | F_{4}, E_{6} | | |
| 0902.0431_FO3140 | 206 | 1.000 | \sigma^{\prime} | | |
| 0902.0431_FO3141 | 206 | 1.000 | \sigma, \gamma | | |
| 0902.0431_FO3142 | 206 | 1.000 | \mathfrak{g}_{e v}, \mathfrak{g}_{0} | | |
| 0902.0431_FO3143 | 206 | 1.000 | G=E_{8} | | |
| 0902.0431_FO3144 | 206 | 1.000 | G^{\sigma, \sigma^{\prime}} | | |
| 0902.0431_FO3145 | 206 | 1.000 | \sigma, \sigma^{\prime} | | |
| 0902.0431_FO3146 | 206 | 0.990 | \left(\mathfrak{e}_{8}\right)^{\sigma, \sigma^{\prime}} | | |
| 0902.0431_FO3147 | 206 | 0.990 | \mathfrak{s o}(8) \oplus | | |
| 0902.0431_FO3148 | 206 | 0.928 | \mathfrak{s o}(8) | | |
| 0902.0431_FO3149 | 206 | 1.000 | \mathfrak{g}_{e d} | | |
| 0902.0431_FO3150 | 206 | 1.000 | G=E_{7} | | |
| 0902.0431_FO3151 | 207 | 0.998 | \left(G_{2}\right)^{C} / S L(3, C) \simeq\left(S^{C}\right)^{6}, G_{2(2)} / S L(3, \boldsymbol{R}) \simeq S_{3,4} | | |
| 0902.0431_FO3152 | 207 | 1.000 | E_{6(6)} | | |
| 0902.0431_FO3153 | 207 | 1.000 | E_{6(-14)} | | |
| 0902.0431_FO3154 | 207 | 1.000 | E_{6(2)} | | |
| 0902.0431_FO3155 | 207 | 1.000 | F_{4,1} | | |
| 0902.0431_FO3156 | 207 | 1.000 | G_{2}{ }^{\prime} | | |
| 0902.0431_FO3157 | 207 | 1.000 | F_{4,2} | | |
| 0902.0431_FO3158 | 208 | 1.000 | { }^{\dagger} | | |
| 0902.0431_FO3159 | 208 | 1.000 | E_{6(-78)} | | |
| 0902.0431_FO3160 | 208 | 1.000 | (S U(8)) / \boldsymbol{Z}_{2} | | |
| 0902.0431_FO3161 | 208 | 1.000 | E_{7(7)} | | |
| 0902.0431_FO3162 | 208 | 0.807 | G, \mathrm{I}, G=G_{2}, F_{4}, E_{6} | | |
| 0902.0431_FO3163 | 208 | 1.000 | G=G_{2}, F_{4} | | |
| 0902.0431_FO3164 | 208 | 0.650 | \mathfrak{g}_{\mathrm{eV}}, \mathfrak{g}_{0} | | |
| 0902.0431_FO3165 | 208 | 0.650 | G=G_{2}, F_{4}, E_{6} | | |
| 0902.0431_FO3166 | 208 | 0.995 | \mathfrak{g}_{\mathrm{ev}}, \mathfrak{g}_{0} | | |
| 0902.0431_FO3167 | 208 | 0.999 | E_{8} / E_{7} | | |
| 0902.0431_FO3168 | 208 | 0.999 | \left((S U(3) \times S U(6)) / Z_{2}\right) . Z_{2} | | |
| 0902.0431_FO3169 | 208 | 1.000 | E_{7, \sigma} | | |
| 0902.0431_FO3170 | 208 | 1.000 | E_{7(-5)} | |