Free products of compact quantum groups
type: Item
instance of: scholarly article
title: Free products of compact quantum groups
zbMATH Open document ID: 0838.46057
DOI: 10.1007/BF02101540
author: Shuzhou Wang
published in: Communications in Mathematical Physics
publication date: 1995-06-22T00:00:00Z
review text: This paper is devoted to the use of free products of \(C^*\)-algebras in the theory of quantum groups (in the sense of Woronowicz). The author first collects basic definitions and results on the category of Woronowicz \(C^*\)-algebras and the dual category of compact quantum groups. Next, he discusses inductive limits of Woronowicz \(C^*\)-algebras. The main results of the paper concern free products of Woronowicz \(C^*\)-algebras. First, it is shown that on the free product of two Woronowicz \(C^*\)-algebras amalgamated over a common Woronowicz \(C^*\)-subalgebra there is a natural structure of a Woronowicz \(C^*\)-algebras, and this is actually the amalgamated free product in the category of Woronowicz \(C^*\)-algebras. Next, the author shows that the Haar state on a free product of Woronowicz \(C^*\)-algebras (amalgamated over the scalars) is the free product of the Haar states of the factors and he describes the irreducible representations of such a free product. As an application of his results the author constructs for each natural number \(m\geq 2\) two compact quantum groups \(A_u(m)\) and \(A_o(m)\) each generated by \(m^2\) elements requiring that the matrix of the generators and its transpose are both unitary respectively orthogonal. He shows that these are pairwise non-isomorphic (even as \(C^*\)-algebras) and that they are not nuclear and thus cannot be isomorphic to \(U_q(m)\) or \(\mathrm{SU}_q(m)\).
Mathematics Subject Classification ID: 46L87
Mathematics Subject Classification ID: 46L85
Mathematics Subject Classification ID: 16T20
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 46M40
zbMATH DE Number: 763766
zbMATH Keywords: free products of \(C^*\)-algebras
zbMATH Keywords: quantum groups
zbMATH Keywords: Woronowicz \(X^*\)-algebras
zbMATH Keywords: dual category of compact quantum groups
zbMATH Keywords: inductive limits
reviewed by: Andreas Cap
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 763766