On duality of algebraic quantum groupoids

TL;DR

This paper develops a comprehensive duality theory for algebraic quantum groupoids, extending existing frameworks and overcoming previous limitations, with applications to weak multiplier Hopf algebras and their morphisms.

Contribution

It introduces a general Pontrjagin duality for quantum groupoids in the algebraic setting, expanding upon and unifying prior duality theories.

Findings

Constructed a duality theory for quantum groupoids extending Van Daele's framework.

Computed duals in several explicit examples.

Showed that morphisms of multiplier Hopf algebroids preserve the antipode.

Abstract

Like quantum groups, quantum groupoids frequently appear in pairs of mutually dual objects. We develop a general Pontrjagin duality theory for quantum groupoids in the algebraic setting that extends Van Daele's duality theory for multiplier Hopf algebras and overcomes the finiteness restrictions of the approach of Kadison, Szlach\'anyi, B\"ohm and Schauenburg. Our construction is based on the integration theory for multiplier Hopf algebroids and yields, as a corollary, a duality theory for weak multiplier Hopf algebras with integrals. We compute the duals in several examples and introduce morphisms of multiplier Hopf algebroids to succinctly describe their structure. Moreover, we show that such morphisms preserve the antipode.