Quantum Groups  

In General > s.a. knots; lie algebra; particle statistics; Poisson-Lie Group; Quantum Algebra.
* Idea: Also known as q-deformed Lie algebras, one version of non-commutative geometry; A quantum group is a manifold M together with a set of maps φi: MM × M, dependent on a complex parameter q, such that multiplication of functions on M is modified to be the sum of the products of f and g evaluated at the various pairs of points, respectively; It is thus non-commutative. [Same as Hopf Algebra?]
* History: Invented largely to provide solutions of the Yang-Baxter equation, and hence solvable models in 2D statistical mechanics and 1D quantum mechanics; They have been hugely successful, although not all Yang-Baxter solutions fit into the framework of quantum groups.
* Deformation parameter q: When q → 1, we recover the regular algebra of functions on M; When q ≠ 1, we can think of the points as being fuzzed out; Important special cases are q = a root of unity.
* Status: 1995, Has been done mostly for Lie groups, where the action of the infinitesimal generators becomes a finite action which defines, starting from the identity, a lattice of points on the group; This "discretization" is at the root of the fact that many calculations which are usually divergent become regularized; Has also been done for Minkowski space, but not for curved spaces.
* And other structure: Can be obtained from simple Lie algebras, and they give polynomial link invariants.
> Online resources: see Wikipedia page.

Physics Applications > s.a. non-extensive statistics [Tsallis entropy].
* Gravitation and spacetime: An example of spacetime having quantum group symmetry is κ-Minkowski.
@ In quantum field theory: Finkelstein MPLA(00)ht [general], ht/00 [SLq(3) fields]; Brouder & Oeckl hp/02, in(04)ht/02 [quantum scalar fields as quantum group]; Marcianò a1003-MG12 [quantum field theory and quantum gravity].
@ Gravitation and spacetime: Bimonte et al NPB(98)ht/97 [and general relativity]; Benedetti PRL(09)-a0811 [fractal dimension].
@ And quantum gravity: Majid in(09)ht/06, ht/06-ch [as frame groups]; > s.a. loop quantum gravity.
@ Gauge theories: Finkelstein ht/99; Mesref IJMPA(05)ht/04 [introduction]; Lewandowski & Okołów JMP(09)-a0810 [generalization of the C* algebra of cylindrical functions on a space of connections].
@ Other: Yakaboylu et al PRL(18)-a1809 [as hidden symmetries of quantum impurities].

References > s.a. modified lorentz group.
@ Intros: Ruíz-Altaba AIP(94)ht/93-ln; Koornwinder in(94)ht [compact]; Jaganathan mp/01-proc; Ritter m.QA/02.
@ Textbooks and reviews: Biederharn IJTP(93); Chari & Pressley 94; Chang PRP(95); Kassel 95; Majid 95; Chaichian & Demichev 96; Gómez et al 96; Etingof & Schiffmann 98; Klimyk & Schmüdgen 98; Jones m.OA/03 [and Yang-Baxter equation]; issue JMP(04)#10; Lusztig 10.
@ General references: Maggiore PRD(94); Ciccoli JGP(99) [representations]; Majid & Pachoł a2006 [classification].
@ Special ones: Goswami mp/01 [non-commutative geometry of SUq(2)]; Dabrowski et al CMP(05) [Dirac operator on SUq(2)].
@ And mathematical structures: Frønsdal qa/95-ln [cohomology]; Sawin BAMS(96)qa/95 [links]; Aschieri & Schupp IJMPA(96) [vector fields].
@ Related topics: Frønsdal LMP(97) [deformations]; Majid JGP(98) [quantum Diff]; Bhowmick & Goswami CMP(09) [quantum isometry groups]; Lehrer et al CMP(11) [associated non-commutative associative algebras, invariants].
> Related topics: see differential geometry [quantum group of isometries].

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