locally compact quantum group

Definition 0.1.

A locally compact quantum group defined as in ref. [1] is a quadruple $\mathrm{G}\mathrm{=}\mathrm{(}A\mathrm{,}\mathrm{\Delta }\mathrm{,}\mu \mathrm{,}\nu \mathrm{)}$, where $A$ is either a $C^{*}$– or a $W^{*}$– algebra equipped with a co-associative comultiplication (http://planetmath.org/WeakHopfCAlgebra2) $\mathrm{\Delta }\mathrm{:}A\mathrm{\to }A\mathrm{\otimes }A$ and two faithful semi-finite normal weights, $\mu$ and $\nu$ –right and -left Haar measures.

0.0.1 Examples

1. An ordinary unimodular group $G$ with Haar measure $\mu$. $A:=L^{\mathrm{\infty }}(G,\mu ),\mathrm{\Delta }:f(g)\mapsto f(gh)$, $S:f(g)\mapsto f(g-1),\varphi (f)={\int }_{G}f(g)𝑑\mu (g)$, where $g,h\in G,f\in L^{\mathrm{\infty }}(G,\mu )$.

2. A:= Ł(G) is the von Neumann algebra generated by left-translations $L_g$ or by left convolutions $L_f:={\int }_{G}f(g)L_g𝑑\mu (g)$ with continuous functions $f(.)\in L^1(G,\mu )\mathrm{\Delta }:\mapsto L_g\otimes L_g\mapsto L_g^{-1},\varphi (f)=f(e)$, where $g\in G$, and e is the unit of G.

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 group
Defines	local quantum symmetry