PlanetPhysics/L Compact Quantum Groups

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Definition 0.1 A locally compact quantum group defined as in ref. 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

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


 * 1) A = \L (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 $$f(\dot) \in L^1(G,\mu) \Delta: \mapsto L_g \otimes L_g \mapsto L_g^{-1}, \phi(f) = f(e) $$, where $$g \in G$$, and $$e$$ is the unit of $$G$$.