Cosemisimple bialgebras and discrete quantum semigroups
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.
scientific article; zbMATH DE number 1104112
scientific article