quantum groups
0.1 Introduction
Definition 0.1.
A quantum group^{} is often defined as the dual of a Hopf algebra^{} or coalgebra. Actually, quantum groups are constructed by employing certain Hopf algebras as “building blocks”, and in the case of finite groups^{} they are obtained from the latter by Fourier transformation^{} of the group elements.
Let us consider next, alternative definitions of quantum groups that indeed possess extended quantum symmetries and algebraic properties distinct from those of Hopf algebras.
0.2 Quantum Groups, Quantum Operator Algebras and Related Symmetries
Definition 0.2.
Quantum groups are defined as locally compact topological groups endowed with a left Haar measure system, and also with at least one internal quantum symmetry, such as the intrinsic spin symmetry represented by either Pauli matrices^{} or the Dirac algebra of observable spin operators.
For additional examples of quantum groups the reader is referred to the last six publications listed in the bibliography.
Remark 0.1.
One can also consider quantum groups as a particular case of quantum groupoids^{} in the limiting case where there is only one symmetry^{} type present in the quantum groupoid.
0.3 Quantum Groups, Paragroups and Operator Algebras in Quantum Theories
Quantum theories^{} adopted a new lease of life post 1955 when von Neumann beautifully reformulated Quantum Mechanics (QM) in the mathematically rigorous context of Hilbert spaces^{} and operator algebras. From a current physics perspective, von Neumann’s approach to quantum mechanics has done however much more: it has not only paved the way to expanding the role of symmetry in physics, as for example with the WignerEckhart theorem and its applications, but also revealed the fundamental importance in quantum physics of the state space^{} geometry of (quantum) operator algebras. Subsequent developments of the quantum operator algebra^{} were aimed at identifying more general quantum symmetries than those defined for example by symmetry groups, groups of unitary operators and Lie groups^{}. Several fruitful quantum algebraic concepts were developed, such as: the Ocneanu paragroupslater found to be represented by Kac–Moody algebras^{}, quantum ‘groups’ represented either as Hopf algebras or locally compact groups with Haar measure, ‘quantum’ groupoids^{} represented as weak Hopf algebras, and so on. The Ocneanu paragroups case is particularly interesting as it can be considered as an extension^{} through quantization of certain finite group symmetries to infinitelydimensional von Neumann type $I{I}_{1}$ factors (subalgebras^{}), and are, in effect, ‘quantized groups’ that can be nicely constructed as Kac algebras; in fact, it was recently shown that a paragroup can be constructed from a crossed product by an outer action of a Kac algebra. This suggests a relation^{} to categorical aspects of paragroups (rigid monoidal tensor categories previously reported in the literature). The strict symmetry of the group of (quantum) unitary operators is thus naturally extended through paragroups to the symmetry of the latter structure^{}’s unitary representations^{}; furthermore, if a subfactor of the von Neumann algebra^{} arises as a crossed product by a finite group action, the paragroup for this subfactor contains a very similar^{} group structure to that of the original finite group, and also has a unitary representation theory similar to that of the original finite group. Lastbutnot least, a paragroup yields a complete^{} invariant^{} for irreducible^{} inclusions of AFD von Neumannn type $I{I}_{1}$ factors with finite index and finite depth (Theorem 2.6. of Sato, 2001). This can be considered as a kind of internal, ‘hidden’ quantum symmetry of von Neumann algebras.
On the other hand, unlike paragroups, (quantum) locally compact groups are not readily constructed as either Kac or Hopf C*algebras. In recent years the techniques of Hopf symmetry and those of weak Hopf C*algebras, sometimes called quantum groupoids (cf Böhm et al.,1999), provide important tools–in addition to the paragroups– for studying the broader relationships of the Wigner fusion rules algebra, $6j$–symmetry (Rehren, 1997), as well as the study of the noncommutative symmetries of subfactors within the Jones tower constructed from finite index depth 2 inclusion of factors, also recently considered from the viewpoint of related Galois correspondences (Nikshych and Vainerman, 2000).
Remark 0.2.
Compact quantum groups (CQGs) (http://planetmath.org/CompactQuantumGroup) are of great interest in physical mathematics especially in relation to locally compact quantum groups (LCQGs) (http://planetmath.org/LocallyCompactQuantumGroup).
References
 1 M. Chaician and A. Demichev: Introduction to Quantum Groups, World Scientific (1996).
 2 V. G. Drinfel’d: Quantum groups, In Proc. Intl. Congress of Mathematicians, Berkeley 1986, (ed. A. Gleason), Berkeley, 798820 (1987).
 3 P.. I. Etingof and A. N. Varchenko, Solutions of the Quantum Dynamical YangBaxter Equation and Dynamical Quantum Groups, Comm.Math.Phys., 196: 591640 (1998).
 4 P. I. Etingof and A. N. Varchenko: Exchange dynamical quantum groups, Commun. Math. Phys. 205 (1): 1952 (1999)
 5 P. I. Etingof and O. Schiffmann: Lectures on the dynamical Yang–Baxter equations, in Quantum Groups and Lie Theory (Durham, 1999), pp. 89129, Cambridge University Press, Cambridge, 2001.
 6 J. M. G. Fell.: The Dual Spaces^{} of C*–Algebras., Transactions of the American Mathematical Society, 94: 365–403 (1960).
 7 P. Hahn: Haar measure for measure groupoids., Trans. Amer. Math. Soc. 242: 1–33(1978).
 8 P. Hahn: The regular representations^{} of measure groupoids., Trans. Amer. Math. Soc. 242:34–72(1978).

9
C. Heunen, N. P. Landsman, B. Spitters.: A topos for algebraic quantum theory, (2008)
arXiv:0709.4364v2 [quant–ph]  10 S. Majid. Quantum groups, http://www.ams.org/notices/200601/whatis.pdfon line
Title  quantum groups 
Canonical name  QuantumGroups 
Date of creation  20130322 18:12:22 
Last modified on  20130322 18:12:22 
Owner  bci1 (20947) 
Last modified by  bci1 (20947) 
Numerical id  38 
Author  bci1 (20947) 
Entry type  Topic 
Classification  msc 81T25 
Classification  msc 81T18 
Classification  msc 81T13 
Classification  msc 81T10 
Classification  msc 81T05 
Classification  msc 81R50 
Synonym  Hopf algebras 
Synonym  locally compact groupoids with Haar measure 
Related topic  HopfAlgebra 
Related topic  HaarMeasure 
Related topic  LocallyCompactQuantumGroup 
Related topic  CompactQuantumGroup 
Related topic  GroupoidAndGroupRepresentationsRelatedToQuantumSymmetries 
Related topic  QuantumGroupoids2 
Related topic  Groupoids 
Related topic  QuantumSpaceTimes 
Related topic  QuantumCategory 
Related topic  LocallyCompactQuantumGroupsUniformContinuity2 
Related topic  UniformCon 
Defines  quantum group 
Defines  local quantum symmetry 