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From Humphreys' Introduction to Lie Algebras and Representation Theory, written in parentheses:

(It can be shown that even dimensionality is a necessary condition for existence of a non-degenerate bilinear form satisfying $f(v,w)=-f(w,v)$.)

Why is this? Suppose we have odd dimensionality. So $n$ is odd. We have an $n\times n$ matrix $S$ with independent rows such that $v^TSw=-w^TSv$ for all vectors $v,w$. How does this provide a contradiction?


marked as duplicate by mdp, Amzoti, Start wearing purple, azimut, Cameron Buie Aug 9 '13 at 13:17

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  • $\begingroup$ It's the non-degeneracy requirement that forces even dimension. $\endgroup$ – Olivier Bégassat Aug 9 '13 at 12:50
  • $\begingroup$ @OlivierBégassat Well, non-degeneracy forces the rows of $S$ to be independent. How does that force even dimension? $\endgroup$ – PJ Miller Aug 9 '13 at 12:52
  • $\begingroup$ My previous comment was made automatically by the software - the linked question is not literally a duplicate, but it contains an answer to this question (which is what the close notification will say if this question is closed as a duplicate). $\endgroup$ – mdp Aug 9 '13 at 12:52

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