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

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Central character of a Lie algebra module

Definition

Let $g$ be a complex Lie algebra and let $M$ be a complex vector space carrying a $g$ -module structure. Let ${\tilde{ρ}}_{M} ⁣ : U (g) \to {End}_{C} (M)$ be the induced action from Lie algebra actions extend to unital actions of the enveloping algebra. A central character of $M$ is a unital complex-algebra homomorphism

$χ ⁣ : Z (U (g)) \to C$

such that every central element $z \in Z (U (g))$ acts by

${\tilde{ρ}}_{M} (z) = χ (z) {id}_{M} .$

When such a map exists, one says that $M$ has central character $χ$ .

Depends on

Used by

Dependency tree · one level

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Sources

Pavel Etingof, Representations of Lie Groups (standard reference, not scraped)
Yiannis Sakellaridis, Verma Modules and the Category O (standard reference, not scraped)