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
*φ*_{i}: *M*
→ *M* × *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

*

*

>

**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
[SL_{q}(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 SU_{q}(2)];
Dabrowski et al CMP(05)
[Dirac operator on SU_{q}(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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