Multiplier Hopf algebroids. Basic theory and examples

TL;DR

This paper develops the foundational theory of multiplier Hopf algebroids, extending quantum groupoid concepts to non-unital and non-separable algebraic contexts, and explores their structural properties and examples.

Contribution

It introduces the basic theory of multiplier Hopf algebroids, including conditions for antipode existence and invertibility, and provides illustrative examples and special cases.

Findings

Bijectivity of canonical maps is equivalent to antipode existence

Conditions for antipode invertibility are discussed

Several examples and special cases are presented

Abstract

Multiplier Hopf algebroids are algebraic versions of quantum groupoids that generalize Hopf algebroids to the non-unital case and weak (multiplier) Hopf algebras to non-separable base algebras. The main structure maps of a multiplier Hopf algebroid are a left and a right comultiplication. We show that bijectivity of two associated canonical maps is equivalent to the existence of an antipode, discuss invertibility of the antipode, and present some examples and special cases.