Bernstein–Gelfand–Gelfand resolution
E934435
The Bernstein–Gelfand–Gelfand resolution is a fundamental construction in representation theory that provides an explicit, exact sequence resolving finite-dimensional representations of semisimple Lie algebras using complexes of Verma modules.
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical construction
ⓘ
resolution in homological algebra ⓘ tool in representation theory ⓘ |
| appliesTo |
finite-dimensional representations of semisimple Lie algebras
ⓘ
semisimple Lie algebras ⓘ |
| assumes |
choice of Borel subalgebra
ⓘ
choice of Cartan subalgebra ⓘ triangular decomposition of a semisimple Lie algebra ⓘ |
| constructs | complexes of Verma modules ⓘ |
| context | category O of Bernstein–Gelfand–Gelfand ⓘ |
| differentialsDependOn | Bruhat order on the Weyl group NERFINISHED ⓘ |
| domain | complex semisimple Lie algebra ⓘ |
| exactAt | all terms except the last ⓘ |
| feature |
Weyl group indexed terms
ⓘ
explicit differentials ⓘ functorial construction ⓘ |
| field |
Lie theory
NERFINISHED
ⓘ
homological algebra ⓘ representation theory ⓘ |
| generalizationOf | resolutions in the sl2 case ⓘ |
| goal | resolve finite-dimensional highest weight modules ⓘ |
| input | irreducible finite-dimensional highest weight module ⓘ |
| inspired | later constructions in geometric representation theory ⓘ |
| lastCohomology | given finite-dimensional module ⓘ |
| moduleType | highest weight modules in category O ⓘ |
| namedAfter |
Israel Gelfand
NERFINISHED
ⓘ
Joseph Bernstein NERFINISHED ⓘ Sergei Gelfand NERFINISHED ⓘ |
| output | exact complex of Verma modules ⓘ |
| property | exact sequence ⓘ |
| relatedTo |
BGG category O
NERFINISHED
ⓘ
BGG reciprocity NERFINISHED ⓘ Borel–Weil–Bott theorem NERFINISHED ⓘ Kazhdan–Lusztig theory NERFINISHED ⓘ Verma module filtration ⓘ Weyl group NERFINISHED ⓘ flag variety cohomology ⓘ highest weight representations ⓘ |
| resolutionType | projective resolution in category O ⓘ |
| termIndexing | elements of the Weyl group ⓘ |
| termStructure | direct sums of Verma modules ⓘ |
| typicalSetting | complex semisimple Lie algebra over C ⓘ |
| usedFor |
computing Ext-groups in category O
ⓘ
computing characters of representations ⓘ deriving BGG reciprocity formulas ⓘ studying composition series of modules ⓘ |
| uses | Verma modules NERFINISHED ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.