Quantum
Groups |

**In General** > s.a. knots;
lie algebra; particle
statistics; Poisson-Lie Group;
Quantum Algebra.

* __Idea__: One version
of non-commutative geometry; 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].

**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.

@ __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__: Maggiore PRD(94);
Frønsdal LMP(97)
[deformations]; Majid
JGP(98)
[quantum Diff]; Ciccoli JGP(99)
[representations]; 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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jul 2016