Quantum Groups| 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: 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 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].
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 2 jul 2020