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Let $gl_S(n,F) = \{x \in gl(n,F) : x^\tau S=-Sx\}$. I have already established that if $P$ is an invertible matrix and $T=P^{\tau}SP$ then $gl_S(n,F) \cong gl_T(n,F)$ by the map $x \rightarrow P^{-1}xP$.

I am now wondering when the converse holds; that is, if $gl_S(n,F) \cong gl_T(n,F)$, then is $S \cong T$ under the equivalence relation of congruence of matrices? More specifically, does it hold over $\mathbb{C}$?

Insights, hints, and complete explanations are all welcome :D

Torsten Schoeneberg
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Eugene
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  • So you are saying that the converse does work?? I'm not seeing how your hint helps, can you please elaborate? Sorry, I'm a low level student!! – Eugene Sep 24 '20 at 12:10
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    I've deleted my earlier comment in which I confused (myself and) you. I've also edited the question to what I think you want, please check. I now think the answer is "yes", but it's not elementary to prove. Jacobson wrote a lot about that in his Lie Algebras book. Will think about it more. – Torsten Schoeneberg Sep 24 '20 at 16:13
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    Actually, I think it's not entirely true in general: Note that for non-zero $S$ and arbitrary $c \in F^\times$, we have $gl_S(n,F) = gl_{cS}(n,F)$, but $S$ and $T:=cS$ are only congruent if $c$ has a square root in $F$. (As a very concrete example, take $S=I_n$, $T=-I_n$ and $F=\mathbb R$.) I think, however, that this is the only issue and $gl_S \simeq gl_T$ implies that $S$ is congruent to some scalar multiple of $T$. Of course over an algebraically closed field like $\mathbb C$ that problem disappears anyway. – Torsten Schoeneberg Sep 25 '20 at 05:48
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    Oh, but you allow arbitrary (not necessarily symmetric) $S$, that might complicate things. Are not e.g. $gl_{\pmatrix{0&1\0&0}}(2, \mathbb C) \simeq gl_{\pmatrix{1&0\0&1}}(2, \mathbb C)$ (namely, the one-dimensional Lie algebra), but the matrices are very incongruent? – Torsten Schoeneberg Sep 25 '20 at 14:53
  • Hm, a lot of great insights as usual, I need to think about them. What if we just think about $F$ as $\mathbb{C}$, are you able to come up with any strong conclusions? – Eugene Sep 25 '20 at 17:59

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