Quantum
Groups| Quantum
Groups |
In General > s.a. knots;
lie algebra; particle
statistics; Poisson-Lie Group.
* Idea: One version
of non-commutative geometry; A manifold M together
with a set of maps 
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 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.
References > s.a. modified
lorentz group.
@ Intros: Ruíz-Altaba ht/93-ln;
Koornwinder ht/94-in
[compact]; Jaganathan mp/01-in;
Ritter m.QA/02;
Majid ht/06-in
[and quantum gravity], ht/06-in
[as frame groups, and quantum gravity].
@ Textbooks and reviews: Biederharn IJTP(93);
Chari & Pressley
94; Chang PRP(95);
Kassel 95; Majid 95; Gómez et al 96; Etingof & Schiffmann
98; Klimyk & Schmüdgen 98; Jones m.OA/03 [and
Yang-Baxter equation]; issue JMP(04)#10.
@ 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 qa/95 [links];
Aschieri & Schupp IJMPA(96)
[vector fields].
@ And quantum field theory / physics: Bimonte et al NPB(98)ht/97 [and
general relativity]; Finkelstein ht/99 [gauge
theories],
MPLA(00)ht [general],
ht/00 [SLq(3)
fields]; Brouder & Oeckl hp/02, ht/02 [quantum
scalar fields as quantum group]; Mesref IJMPA(05)ht/04
[quantum gauge theory, introduction].
@ Related topics: Maggiore PRD(94);
Frønsdal LMP(97)
[deformations]; Majid
JGP(98)
[quantum Diff]; Ciccoli JGP(99)
[representations].
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
11 jul 2008