Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I have the following exercice which I have be trying to solve:

Let L be a Lie algebra and $r:L\rightarrow gl_3(F)$ a representation of L such that $im(r)=t_3(F)$ (the upper triangular matrices). Show that the L-module associated is indecomposable.

Thank you very much for your help!

share|improve this question
    
What is $t_3(F)$? –  Brad Dec 3 '12 at 20:34
    
the upper triangular matrices –  Nre Dec 3 '12 at 20:41

1 Answer 1

up vote 1 down vote accepted

To be explicit, the associated $L$-module is $F^3$, and the action of $L$ is the usual action of $t_3(F)$.

Hint: Suppose that this $L$-module is decomposable, so that $F^3 = V\oplus W$ for nonzero submodules $V$ and $W$. Then one of these submodules, say $V$, is one-dimensional as an $F$-vector space. Being an $L$-submodule of $F^3$, it must be invariant under the action of $t_3(F)$, as is $W$. Use this information to derive a contradiction.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.