TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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The PBW model of a Verma module

Statement

Multiplication gives a vector-space isomorphism

$U (n^{-}) \tilde{\to} M (λ), u ⟼ u v_{λ} .$

Consequently, ordered negative-root PBW monomials from PBW gives an ordered monomial basis for the enveloping algebra applied to $v_{λ}$ form a basis.

Facts & Assumptions

Given: The triangular decomposition and the induced-module relations defining $M (λ)$ .

Proof

technique · direct

1.1

Triangular PBW from Triangular decomposition from a chosen positive root system identifies $U (g)$ with $U (n^{-}) \otimes U (b)$ . Tensoring over $U (b)$ with $C_{λ}$ therefore gives $U (n^{-}) \otimes C_{λ}$ .

givenalgebra

2.1

Under that identification, $u \otimes c_{λ}$ maps to $u v_{λ}$ , so the map is bijective. Applying the ordered PBW basis in the negative factor proves the asserted basis statement.

givenalgebra∎

Depends on

Used by

A two-dimensional A2 Verma weight space Example
The sl2 Verma action in the PBW basis Example
The formal character of a Verma module Proposition
Weights of a Verma module lie below lambda Proposition
Existence and uniqueness of the Shapovalov form Theorem
The Shapovalov determinant formula Theorem

Dependency tree · two levels

9 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, Proposition 25.7 (standard reference, not scraped)