DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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Casimir operator relative to an invariant form

Definition

Let $g$ be finite-dimensional and let $B$ be a nondegenerate, symmetric, invariant bilinear form on it. If $x_{1}, \dots, x_{n}$ is a basis and $x^{1}, \dots, x^{n}$ is its $B$ -dual basis, the Casimir element relative to $B$ is

$Ω_{B} = \sum_{i = 1}^{n} x_{i} x^{i} \in U (g) .$

On a representation $ρ : g \to g l (V)$ , its Casimir operator is

$ρ (Ω_{B}) = \sum_{i = 1}^{n} ρ (x_{i}) ρ (x^{i}) .$

The empty sums give zero when $g = 0$ . Nondegeneracy is exactly what identifies $g$ with its dual and supplies the dual basis. Independence of the chosen basis and commutation with the action are proved in the next lemma.

Depends on

Dependency tree · 2 levels

Sources

Weibel, Lie Algebra Homology and Cohomology, §7.8.8