TITLE: What is the  relation between quantum symmetry  and  quantum groups?
QUESTION [15 upvotes]: 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" ?

REPLY [11 votes]: 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