Integration on algebraic quantum groupoids

TL;DR

This paper develops a theory of integration for algebraic quantum groupoids, extending properties known for quantum groups, and lays the groundwork for duality and operator-algebraic extensions.

Contribution

It introduces a comprehensive integration framework for algebraic quantum groupoids, generalizing key properties from algebraic quantum groups.

Findings

Established faithfulness and uniqueness of integrals

Proved existence of modular elements and automorphisms

Provided foundation for duality and operator-algebraic extensions

Abstract

In this article, we develop a theory of integration on algebraic quantum groupoids in the form of regular multiplier Hopf algebroids, and establish the main properties of integrals obtained by Van Daele for algebraic quantum groups before - faithfulness, uniqueness up to scaling, existence of a modular element and existence of a modular automorphism - for algebraic quantum groupoids under reasonable assumptions. The approach to integration developed in this article forms the basis for the extension of Pontrjagin duality to algebraic quantum groupoids, and for the passage from algebraic quantum groupoids to operator-algebraic completions, which both will be studied in separate articles.