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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!

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

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