Tools for working with multiplier Hopf algebras

TL;DR

This paper develops tools for analyzing multiplier Hopf algebras, focusing on module extension techniques and the covering method, to handle the complexities arising from non-unital algebras and coproducts into multiplier algebras.

Contribution

It introduces a systematic approach to extending modules and explains the covering technique for coproducts in multiplier Hopf algebras, with numerous examples.

Findings

Extended modules provide a broader framework for non-unital algebras.

The covering technique is rigorously explained and illustrated.

Examples demonstrate the applicability of the tools in various contexts.

Abstract

Let $(A,\Delta)$ be a multiplier Hopf algebra. In general, the underlying algebra $A$ need not have an identity and the coproduct $\Delta$ does not map $A$ into $A\otimes A$ but rather into its multiplier algebra $M(A\otimes A)$. In this paper, we study {\it some tools} that are frequently used when dealing with such multiplier Hopf algebras and that are typical for working with algebras without identity in this context. The {\it basic ingredient} is a unital left $A$-module $X$. And the basic construction is that of extending the module by looking at linear maps $\rho:A\to X$ satisfying $\rho(aa')=a\rho(a')$ where $a,a'\in A$. We write the module action as multiplication. Of course, when $x\in X$, and when $\rho(a)=ax$, we get such a linear map. And if $A$ has an identity, all linear maps $\rho$ have this form for $x=\rho(1)$. However, the point is that in the case of a non-unital algebra, the space of such maps is in general strictly bigger than $X$ itself. We get an {\it extended module}, denoted by $\bar X$ (for reasons that will be explained in the paper). We study all sorts of more complicated situations where such extended modules occur and we illustrate all of this with {\it several examples}, from very simple ones to more complex ones where iterated extensions come into play. We refer to cases that appear in the literature. We use this basic idea of extending modules to explain, in a more rigorous way, the so-called {\it covering technique}, which is needed when using {\it Sweedler notations} for coproducts and coactions. Again, we give many examples and refer to the existing literature where this technique is applied.