Subspaces of 7 x 7 skew-symmetric matrices related to the group G_2
Abstract
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 $G_2(K)$.
- Publication:
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arXiv e-prints
- Pub Date:
- November 2008
- DOI:
- 10.48550/arXiv.0811.1298
- arXiv:
- arXiv:0811.1298
- Bibcode:
- 2008arXiv0811.1298G
- Keywords:
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- Mathematics - Group Theory;
- Mathematics - Rings and Algebras;
- 17A36
- E-Print:
- 10 pages