What kind of role do quantum groups play in modern physics ? Do quantum groups naturally arise in quantum mechanics or quantum field theories? What should quantum symmetry refer to ? Can we say that the "symmetry" of a noncommutative space (quantum phase space) should be a quantum group? Do quantum groups describe "extended symmetry" ?
2$\begingroup$ It wouldn't hurt your chances for a better answer if you included some motivation to asking this question/ what have you read so far etc. In my opinion the question as stated now is vague enough to only admit an encyclopedic answer (i.e. nothing you wouldn't find on Wiki). $\endgroup$– Gjergji ZaimiFeb 22, 2010 at 5:54
1$\begingroup$ To follow up on what GZ said: this question has the potential to become a good question, if it's dramatically expanded and motivated. As it is, it's not great. The short answer is that "quantum groups" were invented in the study of quantum integrable systems; they play the role there that Lie groups play in the theory of integrable systems. $\endgroup$– Theo Johnson-FreydFeb 22, 2010 at 16:21
4$\begingroup$ Tensor product is very natural in quantum mechanics since the space of quantum states of a pair of particles is the tensor product of their spaces of states (this is one of the main principles of mathematical modeling of quantum mechanics). This is how tensor products of representations arise in physics. $\endgroup$– Pavel EtingofFeb 23, 2010 at 13:13
2$\begingroup$ @James; it's not appropriate to use the comments to ask more questions. Probably best to pause and think, and write a new question following the guidelines at mathoverflow.net/howtoask. $\endgroup$– Scott MorrisonFeb 24, 2010 at 20:12
1$\begingroup$ @Pavel Didn't you teach a course at MIT several years ago on mathematical methods of physics that emphasize the role of tensor products?I think the notes for that course are still available at your website. $\endgroup$– The MathemagicianJun 23, 2010 at 7:35
Yes, quantum groups naturally arise in many physics problems. E.g. solutions of the quantum Yang-Baxter equation appear as scattering matrices of integrable 2-dimensional quantum field theories (see "Quantum fields and Strings: a course for Mathematicians", p.1179). Also, quantum groups appear in the description of monodromy of the vertex operators in the WZW model of conformal field theory ("the Drinfeld-Kohno theorem"). Thirdly, there are spectacular applications of (infinite dimensional) quantum groups to statistical mechanics, which are described in the book by Jimbo and Miwa "Algebraic analysis of solvable lattice models". Also, quantum groups are useful in construction and studying of certain classes of integrable systems (q-Toda systems, Macdonald-Ruijsenaars systems, etc.)
One of the main mechanisms through which quantum groups appear in physics is the same as for usual Lie groups: if a Hamiltonian of a quantum system has a Lie group symmetry then this helps find its eigenvalues and eigenvectors (which is the main problem in studying a quantum system), because its eigenspaces are representations of this group.
It is not quite true that the usual quantum groups are symmetries of WZNW model. The standard root of unity q. groups are hidden symmetries at the level of pre-Hilbert space where some spurious ghost-like norm zero states appear. The true symmetry (in the sense of axiomatic quantum field theory) includes rather certain quotient which is just a weak quasi-Hopf algebra, and whose representation automatically exclude the nonphysical representations with quantum dimension zero. See
Gerhard Mack, Volker Schomerus: QuasiHopf quantum symmetry in quantum theory. DESY-91-037, May 1991. 37pp. Nucl.Phys.B370:185-230,1992, doi, scan