quantum groups

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 re-formulated 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 Wigner-Eckhart

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

paragroups

-later 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 infinitely-dimensional 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. Last-but-not 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,

6

⁢

j

–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

).

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, 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 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/what-is.pdf

on 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

Generated on Sat Feb 10 13:07:06 2018 by

LaTeXML