Weyl’s theorem on complete reducibility

Theorem (Weyl complete reducibility)

Let $\mathfrak g$ be a finite-dimensional semisimple Lie algebra over a field of characteristic $0$ (in particular over $\mathbb C$). Then every finite-dimensional representation of $\mathfrak g$ is completely reducible : if $W\subseteq V$ is a subrepresentation , there exists a $\mathfrak g$-invariant subspace $W'\subseteq V$ such that

Equivalently, every representation is a direct sum of irreducible representations .

Compact Lie group analogue

If $G$ is a compact Lie group , then every finite-dimensional continuous representation of $G$ on a real or complex vector space admits a $G$-invariant inner product (obtained by averaging), hence is completely reducible. This is a key input to results like the Peter–Weyl theorem .

Context and consequences

Complete reducibility is the representation-theoretic backbone of highest-weight theory: it guarantees semisimplicity of the $\mathfrak h$-action and enables the weight space decomposition , which in turn supports the classification of irreducibles by highest weights . It also interacts with structural criteria for semisimplicity, such as nondegeneracy of the Killing form (compare equivalent characterizations of semisimplicity ).