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locally compact quantum group

(Definition)

Definition 0.1 A locally compact quantum group defined as in ref. [1] is a quadruple $QCG_l =(A, \Delta, \mu, \nu)$ , where $A$ is either a $C^*$ - or a $W^*$ - algebra equipped with a co-associative comultiplication $\Delta: A \to A \otimes A$ and two faithful semi-finite normal weights, $\mu$ and $\nu$ - right and -left Haar measures.

Examples

An ordinary unimodular group 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 $g, h \in G, f \in L^{\infty} (G, \mu)$ .
A = Ł(G) is the von Neumann algebra generated by left-translations or by left convolutions $L_f = \int_G (f(g)L_g d \mu (g))$ with continuous functions $f(\dot) \in L^1(G,\mu) \Delta: \mapsto L_g \otimes L_g \mapsto L_g^{-1}, \phi(f) = f(e)$ , where $g \in G$ , and is the unit of .

Bibliography

1: Leonid Vainerman. 2003.Locally 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 et Co: Berlin.

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