Measured quantum groupoids associated to proper dynamical quantum groups

TL;DR

This paper constructs measured quantum groupoids from algebraic dynamical quantum groups, linking algebraic and operator algebra frameworks, and sets the stage for applications to specific quantum groups like dynamical SU_q(2).

Contribution

It introduces a method to associate measured quantum groupoids to algebraic dynamical quantum groups using operator algebra techniques.

Findings

Constructed fundamental unitaries for dynamical quantum groups.

Lifted invariant integrals to operator algebra level.

Established a framework for associating quantum groupoids to 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 a purely algebraic objects, Hopf C*-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*-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.