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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Observed surface forms (2)

Surface form	Occurrences
Drinfeld–Jimbo quantum group	1
quantum group U_q(sl_2)	1

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 ⓘ

Referenced by (3)

Full triples — surface form annotated when it differs from this entity's canonical label.

XXZ spin chain → isRelatedTo → Drinfeld–Jimbo quantum groups ⓘ

this entity surface form: quantum group U_q(sl_2)

Vladimir Drinfeld → knownFor → Drinfeld–Jimbo quantum groups ⓘ

Yang–Baxter equation → relatedTo → Drinfeld–Jimbo quantum groups ⓘ

this entity surface form: Drinfeld–Jimbo quantum group