Verma module
E581261
A Verma module is a type of highest-weight module over a Lie algebra that is freely generated from a highest-weight vector and plays a central role in the classification of representations of semisimple Lie algebras.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Verma module canonical | 1 |
| Verma modules | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6282520 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Verma module Context triple: [Lie algebra representation, relatedConcept, Verma module]
-
A.
Harish-Chandra isomorphism
The Harish-Chandra isomorphism is a fundamental result in representation theory that identifies the center of the universal enveloping algebra of a semisimple Lie algebra with the algebra of Weyl group–invariant polynomials on a Cartan subalgebra.
-
B.
Harish-Chandra character formula
The Harish-Chandra character formula is a fundamental result in representation theory that gives an explicit expression for the characters of irreducible admissible representations of real reductive Lie groups.
-
C.
Weyl character formula
The Weyl character formula is a fundamental result in representation theory that gives an explicit expression for the characters of irreducible finite-dimensional representations of semisimple Lie algebras and Lie groups.
-
D.
Hecke operators
Hecke operators are algebraic operators acting on modular forms that play a central role in number theory, particularly in understanding congruences, L-functions, and the arithmetic of modular forms.
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E.
Gelfand–Tsetlin basis
The Gelfand–Tsetlin basis is a canonical, combinatorially defined basis for representations of certain Lie algebras and groups, particularly used in the representation theory of GL(n) and related structures.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Verma module Target entity description: A Verma module is a type of highest-weight module over a Lie algebra that is freely generated from a highest-weight vector and plays a central role in the classification of representations of semisimple Lie algebras.
-
A.
Harish-Chandra isomorphism
The Harish-Chandra isomorphism is a fundamental result in representation theory that identifies the center of the universal enveloping algebra of a semisimple Lie algebra with the algebra of Weyl group–invariant polynomials on a Cartan subalgebra.
-
B.
Harish-Chandra character formula
The Harish-Chandra character formula is a fundamental result in representation theory that gives an explicit expression for the characters of irreducible admissible representations of real reductive Lie groups.
-
C.
Weyl character formula
The Weyl character formula is a fundamental result in representation theory that gives an explicit expression for the characters of irreducible finite-dimensional representations of semisimple Lie algebras and Lie groups.
-
D.
Hecke operators
Hecke operators are algebraic operators acting on modular forms that play a central role in number theory, particularly in understanding congruences, L-functions, and the arithmetic of modular forms.
-
E.
Gelfand–Tsetlin basis
The Gelfand–Tsetlin basis is a canonical, combinatorially defined basis for representations of certain Lie algebras and groups, particularly used in the representation theory of GL(n) and related structures.
- F. None of above. chosen
Statements (51)
| Predicate | Object |
|---|---|
| instanceOf |
highest-weight module
ⓘ
mathematical object ⓘ representation theory concept ⓘ |
| builtByActionOf | universal enveloping algebra of the negative nilpotent subalgebra ⓘ |
| constructedFrom |
Borel subalgebra
ⓘ
linked to:
Borel subalgebras
Cartan subalgebra ⓘ highest weight ⓘ triangular decomposition of a Lie algebra ⓘ universal enveloping algebra ⓘ |
| contains | unique (up to scalar) highest-weight vector ⓘ |
| definedOver |
Kac–Moody algebra
ⓘ
linked to:
Kac–Moody algebras
Lie algebra ⓘ semisimple Lie algebra ⓘ |
| field |
Lie theory
ⓘ
algebra ⓘ representation theory ⓘ |
| generalizationOf | highest-weight representations of sl₂ ⓘ |
| hasProperty |
character determined by highest weight
ⓘ
cyclic module ⓘ freely generated from a highest-weight vector ⓘ generated by a highest-weight vector ⓘ graded by weight lattice ⓘ has unique maximal proper submodule ⓘ has unique simple quotient ⓘ indecomposable in general ⓘ typically reducible ⓘ universal highest-weight module ⓘ weight module ⓘ |
| hasSubmoduleStructureDescribedBy |
Weyl group
ⓘ
root system ⓘ |
| introducedInContext | representations of complex semisimple Lie algebras ⓘ |
| isInducedFrom | one-dimensional representation of a Borel subalgebra ⓘ |
| namedAfter | Daya-Nand Verma ⓘ |
| parameterizedBy | highest weight λ ⓘ |
| parameterLivesIn |
dual of Cartan subalgebra
ⓘ
weight lattice ⓘ |
| relatedConcept |
Weyl module
ⓘ
parabolic Verma module ⓘ simple highest-weight module ⓘ |
| restrictionOf | induced module from a one-dimensional Borel representation ⓘ |
| roleIn |
BGG category O
ⓘ
Bernstein–Gelfand–Gelfand reciprocity ⓘ
linked to:
Bernstein–Gelfand–Gelfand resolution
Kazhdan–Lusztig theory ⓘ classification of irreducible representations of semisimple Lie algebras ⓘ highest-weight representation theory of Kac–Moody algebras ⓘ study of primitive ideals in enveloping algebras ⓘ |
| typicalNotation | M(λ) ⓘ |
| universalProperty | initial object among highest-weight modules with fixed highest weight ⓘ |
| usedToCompute | characters of irreducible highest-weight modules ⓘ |
| usedToStudy |
composition series of highest-weight modules
ⓘ
homological properties in category O ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Verma module Description of subject: A Verma module is a type of highest-weight module over a Lie algebra that is freely generated from a highest-weight vector and plays a central role in the classification of representations of semisimple Lie algebras.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Verma module