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

Used by

The Casimir operator is basis-independent and intertwining Lemma
Second Whitehead lemma Theorem
Weyl's complete reducibility theorem Theorem

Dependency tree · two levels

12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Weibel, Lie Algebra Homology and Cohomology, §7.8.8 (standard reference, not scraped)