1

For some standard Lie algebra like $GL_n(R)$, I want to pick some bilinear form $f: g \times g \to R$ that would be anti-symmetric (or alternating): $f(x,y)=-f(y,x)$, maybe defined as $f([x,y])$. Of course there are multiple ways to do that.

Are there examples of such forms that arise naturally in some context, in the same way the Killing form is "natural"?

user2907934
  • 11
  • i'm pretty sure the Lie bracket satisfies this? –  Mar 31 '21 at 14:44
  • 1
    You're right. I should clarify the question -- I was thinking of f() that has values in R – user2907934 Mar 31 '21 at 14:49
  • If the "natural" condition would be that it is invariant wrt the $g$-action (like the Killing form is), then: i. there is a nice theory about non-deg. bil. forms on simple representations $V$ of split semisimple LAs: These are always unique up to scalar, hence either all alternating or all symmetric (Bourbaki, LIE VIII §7 no. 5 prop 12); ii. in particular though, that means that for the adjoint rep -- which is the case you're asking about -- there is no $g$-invariant alternating form, because that space is filled up by scalar multiples of the Killing form, which is symmetric. – Torsten Schoeneberg Apr 01 '21 at 04:38
  • "Of course there are multiple ways to do that": Well as long as we do not make any use of the Lie algebra structure, there are as many as there are alternating forms on a vector space of dimension $d:=\dim g$; i.e. skew-symmetric $d\times d$-matrices. – Torsten Schoeneberg Apr 01 '21 at 04:42

0 Answers0