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 ${\displaystyle {\mathfrak {g}}}$ be a semisimple Lie algebra over a field of characteristic zero. The theorem states that every finite-dimensional module over ${\displaystyle {\mathfrak {g}}}$ 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 ${\displaystyle {\mathfrak {g}}}$ is the complexification of the Lie algebra of a simply connected compact Lie group ${\displaystyle K}$. (If, for example, ${\displaystyle {\mathfrak {g}}=\mathrm {sl} (n;\mathbb {C} )}$, then ${\displaystyle K=\mathrm {SU} (n)}$.) Given a representation ${\displaystyle \pi }$ of ${\displaystyle {\mathfrak {g}}}$ on a vector space ${\displaystyle V,}$ we can first restrict ${\displaystyle \pi }$ to the Lie algebra ${\displaystyle {\mathfrak {k}}}$ of ${\displaystyle K}$. Then, since ${\displaystyle K}$ is simply connected, there is an associated representation ${\displaystyle \Pi }$ of ${\displaystyle K}$. We can then use integration over ${\displaystyle K}$ to produce an inner product on ${\displaystyle V}$ for which ${\displaystyle \Pi }$ is unitary. Complete reducibility of ${\displaystyle \Pi }$ is then immediate and elementary arguments show that the original representation ${\displaystyle \pi }$ of ${\displaystyle {\mathfrak {g}}}$ is also completely reducible.

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