DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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Verma modules

Definition

For $λ \in h^{*}$ , the Verma module is the induced $g$ -module

$M (λ) : = U (g) \otimes_{U (b)} C_{λ},$

where $U (g)$ is the quotient algebra of The universal enveloping algebra as a tensor quotient and $C_{λ}$ is The one-dimensional Borel module of weight lambda. Write $v_{λ} : = 1 \otimes c_{λ}$ . Thus $M (λ)$ is the quotient of $U (g)$ by the left ideal generated by $x$ for $x \in n^{+}$ and $h - λ (h)$ for $h \in h$ ; in particular, $n^{+} v_{λ} = 0$ and $h v_{λ} = λ (h) v_{λ}$ .

Depends on

Used by

A two-dimensional A2 Verma weight space Example
The sl2 Verma action in the PBW basis Example
A proper Verma submodule misses the highest-weight line Lemma
Existence and uniqueness of the Shapovalov form Theorem
The PBW model of a Verma module Theorem
The universal property of Verma modules Theorem

Dependency tree · two levels

3 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

Pavel Etingof, Lie Groups and Lie Algebras I & II, Definition 25.6 (standard reference, not scraped)