How to prove H^2(g,J(g)) is nonzero for a semisimple Lie algebra g, where J(g) is the augmentation ideal of g?

Suppose g is a fiinte dimensional semisimple lie algebra over a field with characteristic 0. This question is related to Whitehead's second lemma, which says for finite dimensional g-module M, H^2(g,M)=0. The solution of this question gives a counterexample to Whitehead's second lemma for infinite dimensional g-module.

edited Dec 11 2011 at 9:16

asked Dec 11 2011 at 9:05

Henry Lee
1●1

I've voted to close as "too localized", as I think this is homework; but even if it isn't, this question is certainly not on the level appropriate for this site. See FAQ for further information. – Leonid Positselski Dec 11 2011 at 9:58

The question makes no sense to me, until the OP makes precise what is the augmentation ideal of a Lie algebra - I know of augmentation ideals for group rings, see en.wikipedia.org/wiki/Augmentation_ideal – Alain Valette Dec 11 2011 at 11:07

Alain- it's the ideal generated by the Lie algebra in the UEA (the kernel of the action on the trivial representation). Every Hopf algebra has an augmentation ideal, which includes the group algebra and UEA examples. – Ben Webster♦ Dec 11 2011 at 14:28

Sounds like homework to me. – Vladimir Dotsenko Dec 11 2011 at 22:23

Browse other questions tagged lie-algebra-cohomology or ask your own question.

question feed