quantum groups

0.1 Introduction

Definition 0.1.

A quantum groupPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath is often defined as the dual of a Hopf algebraPlanetmathPlanetmathPlanetmath or coalgebra. Actually, quantum groups are constructed by employing certain Hopf algebras as “building blocks”, and in the case of finite groupsMathworldPlanetmath they are obtained from the latter by Fourier transformationPlanetmathPlanetmath 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 matricesMathworldPlanetmath 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 groupoidsPlanetmathPlanetmath in the limiting case where there is only one symmetryPlanetmathPlanetmath type present in the quantum groupoid.

0.3 Quantum Groups, Paragroups and Operator Algebras in Quantum Theories

Quantum theoriesPlanetmathPlanetmath adopted a new lease of life post 1955 when von Neumann beautifully re-formulated Quantum Mechanics (QM) in the mathematically rigorous context of Hilbert spacesMathworldPlanetmath 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 Wigner-Eckhart theorem and its applications, but also revealed the fundamental importance in quantum physics of the state spaceMathworldPlanetmath geometry of (quantum) operator algebras. Subsequent developments of the quantum operator algebraPlanetmathPlanetmath were aimed at identifying more general quantum symmetries than those defined for example by symmetry groups, groups of unitary operators and Lie groupsMathworldPlanetmath. Several fruitful quantum algebraic concepts were developed, such as: the Ocneanu paragroups-later found to be represented by Kac–Moody algebrasPlanetmathPlanetmathPlanetmath, quantum ‘groups’ represented either as Hopf algebras or locally compact groups with Haar measure, ‘quantum’ groupoidsPlanetmathPlanetmathPlanetmathPlanetmath represented as weak Hopf algebras, and so on. The Ocneanu paragroups case is particularly interesting as it can be considered as an extensionPlanetmathPlanetmathPlanetmath through quantization of certain finite group symmetries to infinitely-dimensional von Neumann type II1 factors (subalgebrasPlanetmathPlanetmath), 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 relationMathworldPlanetmathPlanetmathPlanetmath 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 structureMathworldPlanetmath’s unitary representationsMathworldPlanetmath; furthermore, if a subfactor of the von Neumann algebraMathworldPlanetmath arises as a crossed product by a finite group action, the paragroup for this subfactor contains a very similarPlanetmathPlanetmath group structure to that of the original finite group, and also has a unitary representation theory similar to that of the original finite group. Last-but-not least, a paragroup yields a completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmath invariantMathworldPlanetmath for irreduciblePlanetmathPlanetmathPlanetmath inclusions of AFD von Neumannn type II1 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 (L-CQGs) (http://planetmath.org/LocallyCompactQuantumGroup).


  • 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, 798-820 (1987).
  • 3 P.. I. Etingof and A. N. Varchenko, Solutions of the Quantum Dynamical Yang-Baxter Equation and Dynamical Quantum Groups, Comm.Math.Phys., 196: 591-640 (1998).
  • 4 P. I. Etingof and A. N. Varchenko: Exchange dynamical quantum groups, Commun. Math. Phys. 205 (1): 19-52 (1999)
  • 5 P. I. Etingof and O. Schiffmann: Lectures on the dynamical Yang–Baxter equations, in Quantum Groups and Lie Theory (Durham, 1999), pp. 89-129, Cambridge University Press, Cambridge, 2001.
  • 6 J. M. G. Fell.: The Dual SpacesMathworldPlanetmathPlanetmathPlanetmath 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 representationsPlanetmathPlanetmath 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/what-is.pdfon line
Title quantum groups
Canonical name QuantumGroups
Date of creation 2013-03-22 18:12:22
Last modified on 2013-03-22 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