Measured quantum transformation groupoids

Abstract

In this article, when $\mathbb{G}$ is a locally compact quantum group, we associate, to a braided-commutative $\mathbb{G}$-Yetter–Drinfel'd algebra $(N,\mathfrak{a},\hat{\mathfrak{a}})$ equipped with a normal faithful semi-finite weight verifying some appropriate condition (in particular if it is invariant with respect to $\mathfrak{a}$, or to $\hat{\mathfrak{a}}$), a structure of a measured quantum groupoid. The dual structure is then given by $(N,\hat{\mathfrak{a}},\mathfrak{a})$. 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.