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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Bernstein–Gelfand–Gelfand resolution canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T11576262 — 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: Bernstein–Gelfand–Gelfand resolution Context triple: [Joseph Bernstein, notableWork, Bernstein–Gelfand–Gelfand resolution]
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A.
Beilinson–Bernstein localization theorem
The Beilinson–Bernstein localization theorem is a fundamental result in geometric representation theory that realizes representations of semisimple Lie algebras as sheaves of differential operators on flag varieties, establishing an equivalence between algebraic and geometric categories.
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B.
Hilbert’s syzygy theorem
Hilbert’s syzygy theorem is a fundamental result in commutative algebra that describes the finite length and structure of free resolutions of modules over polynomial rings.
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C.
Auslander–Buchsbaum formula
The Auslander–Buchsbaum formula is a fundamental result in commutative algebra that relates the projective dimension of a finitely generated module over a Noetherian local ring to the depth of the module and the depth of the ring.
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D.
Kazhdan–Lusztig theory
Kazhdan–Lusztig theory is a framework in representation theory and algebraic geometry that studies Hecke algebras and their bases via Kazhdan–Lusztig polynomials, with deep connections to the representation theory of Lie algebras and geometry of Schubert varieties.
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E.
Borel–Weil theorem
The Borel–Weil theorem is a fundamental result in representation theory that realizes irreducible representations of compact Lie groups as spaces of holomorphic sections of line bundles over their flag manifolds.
- 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: Bernstein–Gelfand–Gelfand resolution Target entity description: 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.
-
A.
Beilinson–Bernstein localization theorem
The Beilinson–Bernstein localization theorem is a fundamental result in geometric representation theory that realizes representations of semisimple Lie algebras as sheaves of differential operators on flag varieties, establishing an equivalence between algebraic and geometric categories.
-
B.
Hilbert’s syzygy theorem
Hilbert’s syzygy theorem is a fundamental result in commutative algebra that describes the finite length and structure of free resolutions of modules over polynomial rings.
-
C.
Auslander–Buchsbaum formula
The Auslander–Buchsbaum formula is a fundamental result in commutative algebra that relates the projective dimension of a finitely generated module over a Noetherian local ring to the depth of the module and the depth of the ring.
-
D.
Kazhdan–Lusztig theory
Kazhdan–Lusztig theory is a framework in representation theory and algebraic geometry that studies Hecke algebras and their bases via Kazhdan–Lusztig polynomials, with deep connections to the representation theory of Lie algebras and geometry of Schubert varieties.
-
E.
Borel–Weil theorem
The Borel–Weil theorem is a fundamental result in representation theory that realizes irreducible representations of compact Lie groups as spaces of holomorphic sections of line bundles over their flag manifolds.
- F. None of above. chosen
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 ⓘ |
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Input
Subject: Bernstein–Gelfand–Gelfand resolution Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.