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Quantum permutation groups: a survey. (English) Zbl 1140.46329

Bozejko, Marek (ed.) et al., Noncommutative harmonic analysis with applications to probability. Papers presented at the 9th workshop, Bȩdlewo, Poland, September 29–October 10, 2006. Warsaw: Polish Academy of Sciences, Institute of Mathematics. Banach Center Publications 78, 13-34 (2008).
Summary: This is a presentation of recent work on quantum permutation groups. It contains: a short introduction to operator algebras and Hopf algebras; quantum permutation groups, and their basic properties; diagrams, integration formulae, asymptotic laws, matrix models; the hyperoctahedral quantum group, free wreath products, quantum automorphism groups of finite graphs, graphs having no quantum symmetry; complex Hadamard matrices, cocycle twists of the symmetric group, quantum groups acting on \(4\) points; remarks and comments.
For the entire collection see [Zbl 1128.46002].

MSC:

46L65 Quantizations, deformations for selfadjoint operator algebras
46L37 Subfactors and their classification
46L54 Free probability and free operator algebras
46L87 Noncommutative differential geometry
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