DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-05

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The quadratic Casimir element

Definition

Let $g$ be a complex semisimple Lie algebra, and let $B$ be the Killing form from The Killing form of a semisimple Lie algebra. Choose dual bases $x_{1}, \dots, x_{n}$ and $x^{1}, \dots, x^{n}$ with respect to $B$ , so $B (x_{i}, x^{j}) = δ_{i}^{j}$ . The quadratic Casimir element is

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

By The quadratic Casimir element is independent of the choice of dual bases ↗, this element is independent of the chosen dual bases.

Depends on

Used by

Dependency tree · one level

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Sources

Pavel Etingof, Representations of Lie Groups (standard reference, not scraped)
Alexander Kleshchev, Lectures on Infinite Dimensional Lie Algebras (standard reference, not scraped)