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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

The BGG resolution has length the number of positive roots Corollary
A nonintegral reflection does not produce a singular power Counterexample
An ambient extension can leave category O Counterexample
The central character does not determine the primitive ideal Counterexample
The full algebraic Verma dual is too large Counterexample
The Soergel functor is not fully faithful on all category-O objects Counterexample
Finite Verma flags and their multiplicities Definition
Parabolic category O for a standard parabolic Definition
Standard and costandard objects Definition
The Bruhat graph and the BGG Verma sum in degree k Definition
The deformed category O family over the Cartan symmetric algebra Definition
The integral parabolic category-O blocks for higher-level Schur-Weyl duality Definition
The regular integral block and its antidominant weight Definition
The singular integral block and its dot stabilizer Definition
The standard induced resolution of the trivial module Definition
Type of a module with a Verma filtration Definition
A point-supported SL2 module gives the shifted Verma module Example
A two-dimensional A2 Verma weight space Example
Primitive ideals of U(sl2) at a generic central character Example
Reducible and generic sl2 Verma modules Example
The annihilator of the trivial sl(2)-module Example
The finite-dimensional sl2 quotient of a Verma module Example
The Kazhdan–Lusztig multiplicity matrix for the sl₃ principal block Example
The regular integral sl2 block Example
The sl2 Verma action in the PBW basis Example
The Soergel functor on the small regular sl₂ block Example
The two projectives in the principal sl2 block Example
The wall-crossing functor on Verma modules in rank one Example
Translation between two regular integral blocks Example
Translation to a singular wall in type A₂ Example
Wall-crossing on the projectives in the sl₂ block Example
A fixed singular Verma chain with globally generated final bundle Lemma
A maximal-label Verma is projective in its truncation Lemma
A proper Verma submodule misses the highest-weight line Lemma
Composition factors of the BGG kernel lie above the degree (BGG 10.6a) Lemma
Exact twisted associated bundles and simple-parabolic descent Lemma
Finite-dimensional tensoring preserves Verma flags Lemma
Finite-dimensional tensoring reaches every simple of a linkage class Lemma
Standard-costandard Hom and Ext-one orthogonality Lemma
The augmentation kernel is the sum of the simple-reflection Verma submodules Lemma

…and 12 more results.

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)