Cosemisimple bialgebras and discrete quantum semigroups
type: Item
instance of: scholarly article
title: Cosemisimple bialgebras and discrete quantum semigroups
zbMATH Open document ID: 0888.16024
author: Eduard Vaysleb
published in: Journal of Algebra
publication date: 1998-03-30T00:00:00Z
review text: A complex bialgebra \(A\) is called cosemisimple [see e.g. section 14 of \textit{M. E. Sweedler}, Hopf algebras. New York: Benjamin (1969; Zbl 0194.32901)]\ if as a coalgebra it is a direct sum of simple coalgebras. A \(*\)-operation (resp. co-\(*\)-operation) on a bialgebra is a conjugate linear involution satisfying certain compatibility properties [see \textit{E. G. Effros} and \textit{Z.-J. Ruan}, Int. J. Math. 5, No. 5, 681--723 (1994; Zbl 0824.17020)]. A \(*\)-operation is said to be standard if for all \(x\in M\), \(x*x=0\) implies that \(x=0\). Effros and Ruan define a discrete quantum group as a cosemisimple Hopf algebra with a standard co-\(*\)-operation; the author thus defines a discrete quantum semigroup to be a cosemisimple bialgebra with a standard co-\(*\)-operation. He shows that for a cosemisimple bialgebra, a standard co-\(*\)-operation making it into a discrete quantum semigroup must be unique. He also proves that such an operation on a cosemisimple \({\mathcal O}_q(\text{SL}_2(\mathbb C))\) exists if and only if the parameter \(q\) is real, and concludes that discrete quantum groups form a more restrictive class than cosemisimple \(*\)-Hopf algebras.
Mathematics Subject Classification ID: 16T10
Mathematics Subject Classification ID: 16T15
Mathematics Subject Classification ID: 16T20
Mathematics Subject Classification ID: 17B37
zbMATH DE Number: 1104112
zbMATH Keywords: Hopf algebras
zbMATH Keywords: quantum groups
zbMATH Keywords: cosemisimple bialgebras
zbMATH Keywords: discrete quantum semigroups
zbMATH Keywords: simple coalgebras
reviewed by: Bethany R. Marsh
MaRDI profile type: Publication
full work available at URL: jabr.1997.7117
OpenAlex ID: W2066314876
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DOI: 10.1006/JABR.1997.7117
description: scientific article; zbMATH DE number 1104112