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Liberation of orthogonal Lie groups. (English) Zbl 1247.46064

This paper pursues the study of the concept of ‘liberation’ of quantum groups, introduced by the first author and others. The authors show that liberation, in the framework of compact orthogonal quantum groups, and under a suitable technical assumption called ‘easiness’, yields a one to one correspondence between classical groups and quantum groups. They classify the groups and show that there are 6 of them.
The notion of liberation for quantum groups is inspired by Voiculescu’s free probability theory: it is an operation that turns a classical compact group into a compact quantum group in such a way that the distribution of the fundamental character is turned into its image under the Bercovici-Pata bijection. Besides it extends the parallels between free and classical probability to the level of combinatorics, via suitable bases of the Tannaka-krein dual, and via asymptotic (free) independence results.

MSC:

46L89 Other “noncommutative” mathematics based on \(C^*\)-algebra theory
20G42 Quantum groups (quantized function algebras) and their representations
22C05 Compact groups
46L54 Free probability and free operator algebras
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