On unitary representation theories of compact quantum groups
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.
compact quantum group-von Neumann algebra
duality between compact and discrete quantum groups
Hopf *-algebra with a characteristic element
scientific article; zbMATH DE number 763759