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.
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
NERFINISHED
ⓘ
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
NERFINISHED
ⓘ
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
NERFINISHED
ⓘ
root system ⓘ |
| introducedInContext | representations of complex semisimple Lie algebras ⓘ |
| isInducedFrom | one-dimensional representation of a Borel subalgebra ⓘ |
| namedAfter | Daya-Nand Verma NERFINISHED ⓘ |
| parameterizedBy | highest weight λ ⓘ |
| parameterLivesIn |
dual of Cartan subalgebra
ⓘ
weight lattice ⓘ |
| relatedConcept |
Weyl module
NERFINISHED
ⓘ
parabolic Verma module ⓘ simple highest-weight module ⓘ |
| restrictionOf | induced module from a one-dimensional Borel representation ⓘ |
| roleIn |
BGG category O
NERFINISHED
ⓘ
Bernstein–Gelfand–Gelfand reciprocity NERFINISHED ⓘ Kazhdan–Lusztig theory NERFINISHED ⓘ 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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.