Yetter-Drinfeld algebras over a pairing of multiplier Hopf algebras

TL;DR

This paper develops a new coaction-based framework for Yetter-Drinfeld algebras over multiplier Hopf algebras, aiming to advance the theory of algebraic quantum transformation groupoids and their self-duality.

Contribution

It introduces an equivalent coaction characterization of Yetter-Drinfeld algebras in the multiplier Hopf algebra setting, extending the standard Hopf algebra approach.

Findings

Established a coaction-based characterization of Yetter-Drinfeld algebras

Applied the framework to Van Daele's algebraic quantum groups

Lays groundwork for self-dual quantum transformation groupoids

Abstract

In the present work, we study Yetter-Drinfeld algebras over a pairing of multiplier Hopf algebras. Our main motivation is the construction of a self-dual theory of (C*-)algebraic quantum transformation groupoids. Instead of the standard characterization of Yetter-Drinfeld algebras given in the case of Hopf algebras, we develop an equivalent "only coaction" characterization in the framework of multiplier Hopf algebras. Finally, as a special case, we focus on Yetter-Drinfeld structures over Van Daele's algebraic quantum groups.