modules for lie algebra

**vernal** · April 6th 2012, 12:21 AM

Hi
let N,L,Q are lie algebras. N is abelian and

an exact sequence.

I want to show that N is Q-MOdule . but i can't. if

$Q\times N\rightarrow N\\ ~~~~~(q,n)~~ \rightarrow ~~~ q.n=\chi ^{-1}(l \chi n l^{-1})\\ ~~~~~~~~with~~~ \pi l=q$

is it true? but i don't show 3 conditions of modules. 3 conditions are:
$1.(\lambda q+\mu q^{'}).n=\lambda (q.n)+\mu (q^{'}.n)\\ ~~~~~~2.~q.(\lambda n+\mu n^{'} )= \lambda (q.n)+\mu (q.n^{'})\\ ~~~~~~3.~ [q,q^{'}].n=q.(q^{'}.n)-q^{'}.(q.n)$

if

$Q\times N\rightarrow N\\ ~~~~~(q,n)~~ \rightarrow ~~~ q.n=\chi ^{-1}(l \chi n l^{-1})\\ ~~~~~~~~with~~~ \pi l=q$

are false, can you tell me q.n=??

thanks.

**vernal** · April 7th 2012, 11:25 AM

Who can help me?

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