The Arabian Journal for Science and Engineering. Section B, Engineering vol:33 issue:2C pages:505-527
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 circle times A but rather into its multiplier algebra M(A circle times A). In this paper, we study 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 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 -> X satisfying rho(aa') = a rho(a') where a, a' is an element of A. We write the module action as multiplication. Of course, when x is an element of 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 extended module, denoted by (X) over bar (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 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 covering technique, which is needed when using Sweedlers notations for coproducts and coactions. Again, we give many examples and refer to the existing literature where this technique is applied.