Question

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?

Answer 1

For finite-dimensional Lie algebras, see e.g. the papers Extensions of Representations of Lie Groups and Lie Algebras, I by G. Hochschild and G.D. Mostow (part II by Mostow is mostly on Lie groups), Extensions of representations of algebraic linear groups by A. Białynicki-Birula and the two preceding authors, and Extensions of representations of Lie algebras by J.G. Ryan. The key words 'extension(s) of representations' will land you a number of other references upon searching in Google or MathSciNet. For loop algebras, which you are interested in, some results can be found in the paper Extensions of modules over loop algebras by Fialowski and Malikov and in this recent preprint by E. Neher and A. Savage.