Re: Invariant bilinear forms on semi-direct products

From: José Carlos Santos (jcsantos_at_fc.up.pt)
Date: 03/24/05 

Date: Thu, 24 Mar 2005 16:00:11 +0000 (UTC)

On 24-03-2005 3:00, deitmar@uni-tuebingen.de wrote:

> No. Consider the following example: g is spanned by one element
> H and V is spanned by one vector X. Assume H.X=X.
> Then the semi-direct product is spanned by H and X and one has
>
> [H,X]=X
>
> Let b be an invariant bilinear form, then
>
> b(X,X) = b(X,[H,X]) = b([X,H],X) = -b(X,X),
>
> so b(X,X)=0. Next
>
> b(H,X) = b(H,[H,X]) = b([H,H],X) = b(0,X) = 0
>
> and finally,
>
> b(X,H) = b([H,X],H) = b(H,[X,H]) = -b(H,X) = 0
>
> So b is degenerate.

Thanks a lot. As a matter of fact, I forgot to add the hypothesis that
in g one can define an invariant non-degenerate bilinear product, but
your answer covers that case too.

What if one alse assumes that g is a simple Lie algebra?

Best regards,

Jose Carlos Santos