# Weyl's theorem on complete reducibility

In algebra, Weyl's theorem on complete reducibility is a fundamental result in the theory of Lie algebra representations. Let ${\mathfrak {g}}$ be a semisimple Lie algebra over a field of characteristic zero. The theorem states that every finite-dimensional module over ${\mathfrak {g}}$ is semisimple as a module (i.e., a direct sum of simple modules.)
Weyl's original proof (for complex semisimple Lie algebras) was analytic in nature: it famously used the unitarian trick. Specifically, one can show that every complex semisimple Lie algebra ${\mathfrak {g}}$ is the complexification of the Lie algebra of a simply connected compact Lie group $K$ . (If, for example, ${\mathfrak {g}}=\mathrm {sl} (n;\mathbb {C} )$ , then $K=\mathrm {SU} (n)$ .) Given a representation $\pi$ of ${\mathfrak {g}}$ on a vector space $V,$ we can first restrict $\pi$ to the Lie algebra ${\mathfrak {k}}$ of $K$ . Then, since $K$ is simply connected, there is an associated representation $\Pi$ of $K$ . We can then use integration over $K$ to produce an inner product on $V$ for which $\Pi$ is unitary. Complete reducibility of $\Pi$ is then immediate and elementary arguments show that the original representation $\pi$ of ${\mathfrak {g}}$ is also completely reducible.
We now explain briefly the role that the quadratic Casimir element $C$ has in the proof. Since $C$ is in the center of the universal enveloping algebra, Schur's lemma tells us that $C$ acts as multiple $c_{\lambda }$ of the identity in the irreducible representation of ${\mathfrak {g}}$ with highest weight $\lambda$ . A key point is to establish that $c_{\lambda }$ is nonzero whenever the representation is nontrivial. This can be done by a general argument  or by the explicit formula for $c_{\lambda }$ . We now consider a very special case of the theorem on complete reducibility: the case where a representation $V$ contains a nontrivial, irreducible, invariant subspace $W$ of codimension one. Let $C_{V}$ denote the action of $C$ on $V$ . Since $V$ is not irreducible, $C_{V}$ is not necessarily a multiple of the identity, but it is a self-intertwining operator for $V$ . Then the restriction of $C_{V}$ to $W$ is a nonzero multiple of the identity. But since the quotient $V/W$ is a one dimensional—and therefore trivial—representation of ${\mathfrak {g}}$ , the action of $C$ on the quotient is trivial. It then easily follows that $C_{V}$ must have a nonzero kernel—and the kernel is an invariant subspace, since $C_{V}$ is a self-intertwiner. The kernel is then a one-dimensional invariant subspace, whose intersection with $W$ is zero. Thus, $\mathrm {ker} (V_{C})$ is an invariant complement to $W$ , so that $V$ decomposes as a direct sum of irreducible subspaces:
$V=W\oplus \mathrm {ker} (C_{V})$ .