# Criterion for Checking When a Lie Algebra Module is Irreducible

Suppose $\mathfrak{g}$ is a complex semisimple Lie algebra and let $V$ be a $\mathfrak{g}$-module. Is there a simple criterion one can use to check whether $V$ is irreducible? The quickest way that I can think of is to look at the dominant weights of $V$, find the highest one, say $\lambda$, and then see if $V$ coincides with the unique irreducible $\mathfrak{g}$-module of highest weight $\lambda$. This seems like a rather brute-force approach though, and I would appreciate if there were an easier way to do this.

All spaces in question are finite-dimensional.

• This depends on how the module has been given. For example, are you able to compute the (formal) character of the module? Commented Dec 15, 2017 at 9:22
• @TobiasKildetoft For the specific example I have in mind, I have the weights given as roots, so I could express the roots in terms of the fundamental weights and then compute the formal character. Commented Dec 15, 2017 at 22:10

## 1 Answer

Look for the subspace of highest-weight vectors, all $v \in V$ satisfying $e_i v = 0$ for all the Chevalley generators $e_i$. The dimension of this space will be the number of irreducible components of $V$.