Drinfeld–Jimbo quantum groups

E884937

Hopf algebra algebraic structure q-deformation quantum group

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.

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All labels observed (6)

Label	Occurrences
Drinfeld–Jimbo quantum groups canonical	2
U_q(sl_2)	2
Drinfeld–Jimbo quantum affine algebras	1
Drinfeld–Jimbo quantum group	1
quantum group U_q(sl_2)	1
quantum groups	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]

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
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 ⓘ linked to: Kac–Moody algebras semisimple Lie algebra ⓘ
contributedBy	Michio Jimbo ⓘ Vladimir Drinfeld ⓘ
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 ⓘ linked to: Chevalley basis
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 ⓘ Vladimir Drinfeld ⓘ
parameterizedBy	q ⓘ
relatedTo	Drinfeld double ⓘ Yangian ⓘ linked to: Yangians quantized universal enveloping algebra ⓘ quantum affine algebra ⓘ
satisfies	q-Serre relations ⓘ quantum Yang–Baxter equation ⓘ linked to: Yang–Baxter equation
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