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The universal property of Verma modules
Statement
For a -module , sending a homomorphism to is a bijection onto the vectors of weight annihilated by . Here is Verma modules. The nonzero vectors in this target are precisely the highest-weight vectors of weight from Highest-weight vectors and cyclic highest-weight modules; the zero vector corresponds to the zero homomorphism.
Facts & Assumptions
Given: A -module and with and .
Proof
Define . If , then , exactly the scalar by which acts on ; hence and descends to the induced module.
The descended map is -linear and sends to . Conversely a homomorphism is determined by , since that vector generates ; its image necessarily has the two displayed properties.
Depends on
Used by
- Equal central character does not give every Verma embedding direction Counterexample
- Ordinary Weyl orbits do not classify central characters Counterexample
- The singular integral block and its dot stabilizer Definition
- Dot-conjugate type-A₂ weights have the same central character Example
- The A2 regular integral-dominant Verma embedding poset Example
- The sl2 Verma embedding chain Example
- The two projectives in the principal sl2 block Example
- A maximal-label Verma is projective in its truncation Lemma
- Dominant integral dot translates embed canonically in the Verma module Lemma
- Finite filtrations by highest-weight quotients Lemma
- Finite-dimensional tensoring reaches every simple of a linkage class Lemma
- Induced modules from finite-dimensional B-modules have type their weights Lemma
- Peeling a maximal-weight Verma from a standard filtration Lemma
- Preliminary factorization of a Shapovalov determinant Lemma
- Restricted self-duality of simple highest-weight modules Lemma
- Simple extensions cannot cross linkage classes Lemma
- Standard-costandard Hom and Ext-one orthogonality Lemma
- Tensoring a Verma module by a finite-dimensional module shifts the type Lemma
- The augmentation kernel is the sum of the simple-reflection Verma submodules Lemma
- The generic radical on a Shapovalov factor hyperplane Lemma
- Verma self-extensions in O split Lemma
- Wall-crossing compositions produce the projective with the top standard label Lemma
- Finite-dimensional Hom spaces in O Proposition
- Simple-reflection embeddings of Verma modules Proposition
- Verma homomorphisms and singular vectors Proposition
- Duflo–Joseph g-finite endomorphisms of every Verma module Theorem
- Lifting of indecomposable projectives to the deformed family Theorem
- Parabolic category O is a highest weight category Theorem
- The simple objects of O Theorem
- Verma embedding for an arbitrary positive root Theorem
Dependency tree · two levels
5 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 25.10 (standard reference, not scraped)