locally compact quantum group

Definition 0.1.

A locally compact quantum groupPlanetmathPlanetmath defined as in ref. [1] is a quadruple G=(A,Δ,μ,ν), where A is either a C*– or a W*algebraPlanetmathPlanetmathPlanetmath equipped with a co-associative comultiplication (http://planetmath.org/WeakHopfCAlgebra2) Δ:AAA and two faithfulPlanetmathPlanetmathPlanetmath semi-finite normal weights, μ and νright and -left Haar measures.

0.0.1 Examples

  1. 1.

    An ordinary unimodular groupMathworldPlanetmathPlanetmath G with Haar measure μ. A:=L(G,μ),Δ:f(g)f(gh), S:f(g)f(g-1),ϕ(f)=Gf(g)𝑑μ(g), where g,hG,fL(G,μ).

  2. 2.

    A:= Ł(G) is the von Neumann algebraMathworldPlanetmathPlanetmathPlanetmath generated by left-translations Lg or by left convolutions Lf:=Gf(g)Lg𝑑μ(g) with continuous functions f(.)L1(G,μ)Δ:LgLgLg-1,ϕ(f)=f(e), where gG, and e is the unit of G.


  • 1 Leonid Vainerman. 2003. http://planetmath.org/?op=getobj&from=papers&id=471Locally Compact Quantum Groups and GroupoidsPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath: Proceedings of the Meeting of Theoretical Physicists and Mathematicians, Strasbourg, February 21-23, 2002., Series in Mathematics and Theoretical Physics, 2, Series ed. V. Turaev., Walter de Gruyter Gmbh & Co: Berlin.
Title locally compact quantum group
Canonical name LocallyCompactQuantumGroup
Date of creation 2013-03-22 18:21:24
Last modified on 2013-03-22 18:21:24
Owner bci1 (20947)
Last modified by bci1 (20947)
Numerical id 18
Author bci1 (20947)
Entry type Definition
Classification msc 81R50
Classification msc 46M20
Classification msc 18B40
Classification msc 22A22
Classification msc 17B37
Classification msc 46L05
Classification msc 22D25
Synonym Hopf algebras
Synonym ring groups
Related topic CompactQuantumGroup
Related topic LocallyCompactQuantumGroupsUniformContinuity2
Related topic RepresentationsOfLocallyCompactGroupoids
Related topic VonNeumannAlgebra
Related topic WeakHopfCAlgebra2
Related topic LocallyCompactHausdorffSpace
Related topic QuantumGroups
Defines quantum groupPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath
Defines local quantum symmetry