Completely reducible representation

Let $\rho$ be a finite-dimensional representation, either of a Lie algebra $\mathfrak{g}$ (a Lie algebra representation ) or of a Lie group $G$ (a Lie group representation ).

Definition. $\rho$ is completely reducible if for every invariant subspace $W\subset V$ (a subrepresentation ), there exists an invariant complement $W'\subset V$ such that

as representations.

Equivalently, $V$ can be written as a finite direct sum of irreducible subrepresentations.

Context and key theorems.

• If $\mathfrak{g}$ is semisimple over $\mathbb{C}$ (or $\mathbb{R}$ with suitable hypotheses), then every finite-dimensional representation is completely reducible; this is Weyl’s complete reducibility theorem .
• If $G$ is compact , then every finite-dimensional continuous representation is completely reducible, via averaging an inner product (a Lie-group analogue of Maschke’s theorem), and more globally by Peter–Weyl .

Complete reducibility is the structural reason highest-weight classifications work for compact and semisimple settings.