# locally compact quantum group

###### Definition 0.1.

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

## 0.0.1 Examples

1. 1.

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

2. 2.

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

## References

• 1 Leonid Vainerman. 2003. http://planetmath.org/?op=getobj&from=papers&id=471Locally Compact Quantum Groups and Groupoids: 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 group Defines local quantum symmetry