Uniqueness of decomposition of $\mathfrak{sl}(2,\mathbb{C})$-modules

Question

By Weyl's Theorem, I know that every $\mathfrak{sl}(2,\mathbb{C})$-module is completely reducible. I'm under the impression that, up to isomorphism, this decomposition is unique, and I would go about proving this by adapting a proof from the representation theory of finite groups. However, given that $\mathfrak{sl}(2,\mathbb{C})$ has a simple, well-understood structure, I was wondering if there existed a simpler proof in this case. If so, could anyone point me in the right direction or give hints as to how to proceed?

Answer 1

The structure of an irreducible $\mathfrak{sl}(2,\mathbb{C})$-module is determined by the eigenvalues of $h =\begin{pmatrix}1 & 0 \\ 0 & -1 \end{pmatrix}$. For finite-dimensional $\mathfrak{sl}(2,\mathbb{C})$-modules in general, the structure is determined by the eigenspaces of $h$.

See chapter 8 of "Introduction to Lie Algebras" by Erdmann and Wildon, also my answer here.