# PlanetPhysics/L Compact Quantum Groups

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Definition 0.1


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

Examples

1. An ordinary unimodular group ${\displaystyle G}$ with Haar measure ${\displaystyle \mu }$:

${\displaystyle A=L^{\infty }(G,\mu ),\Delta :f(g)\mapsto f(gh)}$, ${\displaystyle S:f(g)\mapsto f(g^{}-1),\phi (f)=\int _{G}f(g)d\mu (g)}$, where ${\displaystyle g,h\in G,f\in L^{\infty }(G,\mu )}$.

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

[1]

## References

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.