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Compact quantum groups. (English) Zbl 0997.46045

Connes, A. (ed.) et al., Quantum symmetries/ Symétries quantiques. Proceedings of the Les Houches summer school, Session LXIV, Les Houches, France, August 1 - September 8, 1995. Amsterdam: North-Holland. 845-884 (1998).
The author reviews established results and recent developments in the theory of compact quantum groups. The paper begins with the definition of a compact quantum group as a \(C^*\)-algebra with additional structure provided by the coproduct. The existence of a Haar measure for a compact quantum group is proven, and the theory of unitary representations is discussed. Particular attention is paid to the right regular representation. Next, it is proven that any compact quantum group contains a dense subalgebra which is a \(*\)-Hopf algebra. Finally some elements of the Peter-Weyl theory for compact quantum groups are described, and the structure of compact quantum groups with a faithful Haar measure is discussed.
This is a very clearly written paper that will be of great use to all mathematicians and mathematical physicists who would like to learn quickly about basic key features of compact quantum groups. The reviewer also believes that the paper will be of great service to all researchers who would like to understand the connection between compact quantum groups and the algebraic approach to quantum groups via the theory of Hopf algebras.
For the entire collection see [Zbl 0902.00046].

MSC:

46L89 Other “noncommutative” mathematics based on \(C^*\)-algebra theory
17B37 Quantum groups (quantized enveloping algebras) and related deformations
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
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