Multiplier Hopf algebroids arising from weak multiplier Hopf algebras

TL;DR

This paper explores the relationship between weak multiplier Hopf algebras and multiplier Hopf algebroids, establishing conditions under which one structure arises from the other and providing illustrative examples.

Contribution

It demonstrates that regular weak multiplier Hopf algebras naturally induce regular multiplier Hopf algebroids and characterizes when the reverse construction is possible.

Findings

Regular weak multiplier Hopf algebras induce regular multiplier Hopf algebroids.

Characterization of when a multiplier Hopf algebroid arises from a weak multiplier Hopf algebra.

Examples of multiplier Hopf algebroids not originating from weak multiplier Hopf algebras.

Abstract

It is well-known that any weak Hopf algebra gives rise to a Hopf algebroid. Moreover it is possible to characterize those Hopf algebroids that arise in this way. Recently, the notion of a weak Hopf algebra has been extended to the case of algebras without identity. This led to the theory of weak multiplier Hopf algebras. Similarly also the theory of Hopf algebroids was recently developed for algebras without identity. They are called multiplier Hopf algebroids. Then it is quite natural to investigate the expected link between weak multiplier Hopf algebras and multiplier Hopf algebroids. This relation has been considered already in the original paper on multiplier Hopf algebroids. In this note, we investigate the connection further. First we show that any regular weak multiplier Hopf algebra gives rise, in a natural way, to a regular multiplier Hopf algebroid. Secondly we give a characterization, mainly in terms of the base algebra, for a regular multiplier Hopf algebroid to have an underlying weak multiplier Hopf algebra. We illustrate this result with some examples. In particular, we give examples of multiplier Hopf algebroids that do not arise from a weak multiplier Hopf algebra.