Measured quantum groupoids associated to proper dynamical quantum groups

Abstract

Dynamical quantum groups were introduced by Etingof and Varchenko in connection with the dynamical quantum Yang–Baxter equation, and measured quantum groupoids were introduced by Enock, Lesieur and Vallin in their study of inclusions of type II${}_1$ factors. In this article, we associate to suitable dynamical quantum groups, which are purely algebraic objects, Hopf $C\ast$-bimodules and measured quantum groupoids on the level of von Neumann algebras. Assuming invariant integrals on the dynamical quantum group, we first construct a fundamental unitary which yields Hopf bimodules on the level of $C\ast$-algebras and von Neumann algebras. Next, we assume properness of the dynamical quantum group and lift the integrals to the operator algebras. In a subsequent article, this construction shall be applied to the dynamical SU${}_q$(2) studied by Koelink and Rosengren.