Free products of compact quantum groups
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 \(\mathrm{SU}_q(m)\).
scientific article; zbMATH DE number 763766
scientific article