Complete Reducibility and Weyl's Theorem

Weyl's Theorem establishes complete reducibility for semisimple Lie algebras, analogous to Maschke's theorem for finite groups.

TheoremWeyl's Theorem on Complete Reducibility

Every finite-dimensional representation of a semisimple Lie algebra $\mathfrak{g}$ over a field of characteristic zero is completely reducible.

Equivalently, every subrepresentation has an invariant complement.

Proof strategy: Use a Casimir element—an element in the center of $U(\mathfrak{g})$ that acts as a scalar on irreducibles by Schur's Lemma. Averaging constructions (analogous to Maschke) don't work directly for Lie algebras, so the Casimir provides the key tool.

DefinitionSemisimple Lie Algebra

A Lie algebra $\mathfrak{g}$ is semisimple if it has no non-zero abelian ideals. Equivalently:

The Killing form $B(X,Y) = \text{tr}(\text{ad}(X) \circ \text{ad}(Y))$ is non-degenerate
$\mathfrak{g}$ is a direct sum of simple Lie algebras
The radical (maximal solvable ideal) is zero

ExampleSemisimple Lie Algebras

Simple examples:

$\mathfrak{sl}_n(\mathbb{C})$ for $n \geq 2$
$\mathfrak{so}_n(\mathbb{C})$ for $n \geq 3$ (with exceptions)
$\mathfrak{sp}_{2n}(\mathbb{C})$ for $n \geq 1$

Not semisimple:

$\mathfrak{gl}_n(\mathbb{C}) = \mathfrak{sl}_n \oplus \mathbb{C} \cdot I$ (has center)
Heisenberg algebra (nilpotent)
Borel subalgebras (upper triangular matrices, solvable)

TheoremCasimir Element

For a semisimple Lie algebra $\mathfrak{g}$ with non-degenerate Killing form, choose a basis $\{X_i\}$ and dual basis $\{X^i\}$ (via the Killing form). The Casimir element is: $C = \sum_i X_i X^i \in U(\mathfrak{g})$

Properties:

$C$ is in the center of $U(\mathfrak{g})$ : $[C, X] = 0$ for all $X \in \mathfrak{g}$
On an irreducible representation $V$ , $C$ acts as a scalar $c_V \cdot \text{id}_V$ (by Schur's Lemma)
The scalar $c_V$ can be computed from the highest weight

Remark

Weyl's theorem fails in positive characteristic, leading to modular Lie algebra theory with rich structure involving restricted Lie algebras, Frobenius kernels, and quantum groups at roots of unity.

The theorem is crucial for:

Classification: Reducing the problem to irreducible representations
Characters: Weyl character formula for irreducibles
Tensor products: Clebsch-Gordan decomposition
Physics: Particle multiplets in quantum mechanics correspond to irreducible representations

Complete reducibility connects Lie theory to geometric representation theory (perverse sheaves, D-modules) and harmonic analysis (unitary representations of compact Lie groups).