# Extending Lie algebra homomorphisms.

Let $\frak g$ be a Lie algebra over a field of characteristic zero with a Lie subalgebra $\frak s$ and consider $M$ a $\frak s$-module. When is it possible to extend the action of $\frak s$ to $\frak g$, i.e give a $\frak g$-module structure to $M$?

Notice that I stated $\frak g$ without dimensional restrictions. However, my problem is specifically working with $\frak g \otimes \mathbb C [ t,t^{-1}]$, where $\frak g$ is a finite dimensional Lie algebra over $\mathbb C$.

Here I don't care how the action is extended, I am only interested in existencial stuff.

What could be a good reference to dealt with this problem?

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Any book on the representation theory of Lie algebras would persuade you that the answer is no in general. –  Simon Wadsley Dec 7 '11 at 17:23
@Simon: I will edit the question to avoid further problems. –  Chris Dec 7 '11 at 17:27