Keep It Irreducible!

I discussed that any finite-dimensional Lie algebra admits a decomposition into a semi-direct product of a semisimple Lie subalgebra and a solvable ideal (Levi decomposition). Thus, it is reasonable to work with semisimple Lie algebras, and indeed they happen to have some nice properties. This blog post explores one of the crucial properties of a complex finite-dimensional semisimple Lie algebra.

Recall that a representation of a Lie algebra \(\mathfrak{g}\) is a Lie algebra homomorphism \(\phi:\mathfrak{g}\to\text{End}(V)\) for some vector space \(V\). Then if \(\mathfrak{g}\) is a semisimple Lie algebra, one has the following theorem attributed to Weyl:

Theorem Every finite-dimensional representation of \(\mathfrak{g}\) is completely reducible.

That is, every \(\mathfrak{g}\)-module decomposes into a direct sum of irreducible \(\mathfrak{g}\)-modules. Note the similarity with the setting of representations for a finite group.

Proof