Drinfeld–Jimbo quantum groups
E884937
Drinfeld–Jimbo quantum groups are deformations of universal enveloping algebras of Lie algebras that provide a foundational algebraic framework for quantum integrable systems and modern representation theory.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Drinfeld–Jimbo quantum group | 1 |
| Drinfeld–Jimbo quantum groups canonical | 1 |
| quantum group U_q(sl_2) | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10773432 — 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: Drinfeld–Jimbo quantum groups Context triple: [Vladimir Drinfeld, knownFor, Drinfeld–Jimbo quantum groups]
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A.
Schur–Weyl duality
Schur–Weyl duality is a fundamental result in representation theory that links representations of the symmetric group and the general linear group via their commuting actions on tensor powers of a vector space.
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B.
Yang–Baxter equation
The Yang–Baxter equation is a fundamental consistency condition in mathematical physics and integrable systems that underlies exactly solvable models, quantum groups, and braid group representations.
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C.
affine Lie algebras
Affine Lie algebras are infinite-dimensional extensions of finite-dimensional simple Lie algebras that play a central role in representation theory, conformal field theory, and the study of exactly solvable models in mathematical physics.
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D.
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.
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E.
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.
- 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: Drinfeld–Jimbo quantum groups Target entity description: Drinfeld–Jimbo quantum groups are deformations of universal enveloping algebras of Lie algebras that provide a foundational algebraic framework for quantum integrable systems and modern representation theory.
-
A.
Schur–Weyl duality
Schur–Weyl duality is a fundamental result in representation theory that links representations of the symmetric group and the general linear group via their commuting actions on tensor powers of a vector space.
-
B.
Yang–Baxter equation
The Yang–Baxter equation is a fundamental consistency condition in mathematical physics and integrable systems that underlies exactly solvable models, quantum groups, and braid group representations.
-
C.
affine Lie algebras
Affine Lie algebras are infinite-dimensional extensions of finite-dimensional simple Lie algebras that play a central role in representation theory, conformal field theory, and the study of exactly solvable models in mathematical physics.
-
D.
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.
-
E.
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.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
Hopf algebra
ⓘ
algebraic structure ⓘ q-deformation ⓘ quantum group ⓘ |
| basedOn |
Kac–Moody algebra
NERFINISHED
ⓘ
semisimple Lie algebra ⓘ |
| contributedBy |
Michio Jimbo
NERFINISHED
ⓘ
Vladimir Drinfeld NERFINISHED ⓘ |
| definedOver | field of rational functions in q ⓘ |
| developedIn | 1980s ⓘ |
| field | mathematics ⓘ |
| generalizationOf | universal enveloping algebra ⓘ |
| hasApplication |
construction of braid group representations
ⓘ
construction of quantum knot invariants ⓘ modular tensor categories ⓘ tensor category theory ⓘ |
| hasGeneratorType | Chevalley generators NERFINISHED ⓘ |
| hasProperty |
braided tensor category of representations
ⓘ
quasitriangular Hopf algebra ⓘ rigid monoidal category of representations ⓘ |
| hasRepresentationTheory |
finite-dimensional representations
ⓘ
highest weight representations ⓘ integrable representations ⓘ |
| hasStructure |
R-matrix
ⓘ
antipode ⓘ coproduct ⓘ counit ⓘ |
| introducedInContextOf | quantum integrable systems ⓘ |
| isADeformationOf | universal enveloping algebra of a Lie algebra ⓘ |
| namedAfter |
Michio Jimbo
NERFINISHED
ⓘ
Vladimir Drinfeld NERFINISHED ⓘ |
| parameterizedBy | q ⓘ |
| relatedTo |
Drinfeld double
NERFINISHED
ⓘ
Yangian NERFINISHED ⓘ quantized universal enveloping algebra ⓘ quantum affine algebra ⓘ |
| satisfies |
q-Serre relations
ⓘ
quantum Yang–Baxter equation NERFINISHED ⓘ |
| specializesTo | universal enveloping algebra when q → 1 ⓘ |
| subfield |
mathematical physics
ⓘ
quantum algebra ⓘ representation theory ⓘ |
| usedIn |
conformal field theory
ⓘ
knot invariants ⓘ low-dimensional topology ⓘ quantum integrable systems ⓘ quantum inverse scattering method ⓘ statistical mechanics ⓘ |
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: Drinfeld–Jimbo quantum groups Description of subject: Drinfeld–Jimbo quantum groups are deformations of universal enveloping algebras of Lie algebras that provide a foundational algebraic framework for quantum integrable systems and modern representation theory.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Drinfeld–Jimbo quantum group
this entity surface form:
quantum group U_q(sl_2)