Measured quantum transformation groupoids

TL;DR

This paper constructs measured quantum groupoids from braided-commutative G-Yetter-Drinfel'd algebras associated with locally compact quantum groups, generalizing transformation groupoids and providing new examples.

Contribution

It introduces a novel method to associate measured quantum groupoids to certain algebraic structures related to quantum groups, extending existing frameworks.

Findings

Construction of measured quantum groupoids from braided-commutative G-Yetter-Drinfel'd algebras

Dual structures are explicitly characterized

Examples include quotient type co-ideals of compact quantum groups

Abstract

In this article, when G is a locally compact quantum group, we associate to a braided-commutative G-Yetter-Drinfel'd algebra $(N,a,\hat{a})$ equipped with a normal faithful semi-finite weight verifying some appropriate condition, a structure of a measured quantum groupoid. The dual structure is then given by $(N,\hat{a},a)$. Examples are given, especially the situation of a quotient type co-ideal of a compact quantum group. This construction generalizes the standard construction of a transformation groupoid. Most of the results were announced by the second author in 2011, at a conference in Warsaw.