On unitary representation theories of compact quantum groups
type: Item
instance of: scholarly article
title: On unitary representation theories of compact quantum groups
zbMATH Open document ID: 0830.46065
DOI: 10.1007/BF02101533
author: Shigeru Yamagami
published in: Communications in Mathematical Physics
publication date: 1995-06-22T00:00:00Z
review text: The paper deals with the duality between compact and discrete quantum groups. The main definition in the paper is that of a `compact quantum group-von Neumann algebra'. It is a von Neumann direct sum of finite- dimensional matrix algebras over \(\mathbb{C}\), together with a comultiplication, admitting a counit and an antipode (with a polar decomposition). This notion is essentially the same as the one of discrete quantum groups as defined in [\textit{A. van Daele}, Discrete quantum groups. Preprint K. U. Leuven (1993), to appear in J. Algebra, see also \textit{E. G. Effros} and \textit{Z.-J.Ruan}, Int. J. Math. 5, 681-723 (1994; Zbl 0824.17020)]. The author also introduces the notion of a Hopf *-algebra with a characteristic element. Roughly speaking, it is an element \(h\) such that the square \(h^2\) implements the square of the antipode. For such Hopf *-algebras, it is shown that the *-monoidal category of finite- dimensional unitary *-representations is a so called `unitary representation theory'. To such a unitary representation theory can be associated, in a unique way, a compact quantum group-von Neumann algebra. This result is essentially the Tannaka-Krein duality [see also \textit{S. L. Woronowicz}, Invent. Math. 93, 35-76 (1988; Zbl 0664.58044) and \textit{S. Wang}, Krein duality for compact quantum groups (Preprint K. U. Leuven 1994)]. In the last section of the paper, a proof is given of the existence of left and right invariant Haar measures on compact quantum group-von Neumann algebras.
Mathematics Subject Classification ID: 46L89
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 18D10
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 46K05
Mathematics Subject Classification ID: 46L51
Mathematics Subject Classification ID: 46L53
Mathematics Subject Classification ID: 46L54
Mathematics Subject Classification ID: 46L85
Mathematics Subject Classification ID: 46L87
Mathematics Subject Classification ID: 47L50
zbMATH DE Number: 763759
zbMATH Keywords: compact quantum group-von Neumann algebra
zbMATH Keywords: duality between compact and discrete quantum groups
zbMATH Keywords: Hopf *-algebra with a characteristic element
zbMATH Keywords: *- monoidal category
zbMATH Keywords: unitary *-representations
zbMATH Keywords: Tannaka-Krein duality
reviewed by: Alfons Van Daele
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 763759