与群G_2有关的7 × 7 skew-symmetric matrices的子空间
Subspaces of 7 x 7 skew-symmetric matrices related to the group G_2
摘要：
设$K$是特征不等于2的域，$C$是$K$上的八元代数.证明了$K$上的$7\times 7$反对称矩阵存在一个在$C$的自同构群下不变的7维子空间.当$C$是除代数时，该子空间由秩6的元素组成，当$C$是分裂代数时，该子空间由秩4和秩6的元素组成。在后一种情况下，自同构群是例外群$G_2(K)$。
英文摘要：
Let $K$ be a field of characteristic different from 2 and let $C$ be an octonion algebra over $K$. We show that there is a seven-dimensional subspace of $7\times 7$ skew-symmetric matrices over $K$ which is invariant under the automorphism group of $C$. This subspace consists of elements of rank 6 when $C$ is a division algebra, and elements of rank 4 and 6 when $C$ is a split algebra. In the latter case, the automorphism group is the exceptional group $G_2(K)$.
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