Let G be a finite group. Consider the algebra A of all complex functions on G (with pointwise product). Define a coproduct Δ on A by Δ(f)(p,q) = f(pq) where f ∈ A and p,q ∈ G. Then (A,Δ) is a Hopf algebra. If G is only a groupoid, so that the product of two elements is not always defined, one still can consider A and define Δ(f)(p,q) as above when pq is defined. If we let Δ(f)(p,q) = 0 otherwise, we still get a coproduct on A, but Δ(1) will no longer be the identity in A ⊗ A. The pair (A,Δ) is not a Hopf algebra but a weak Hopf algebra. If G is a group, but no longer finite, one takes for A the algebra of functions with finite support. Then A has no identity and (A,Δ) is not a Hopf algebra but a multiplier Hopf algebra. Finally, if G is a groupoid, but not necessarily finite, the standard construction above will give what we call in this paper a weak multiplier Hopf algebra.
Indeed, this paper is devoted to the development of this 'missing link': weak multiplier Hopf algebras. We denote a great part of this paper to the motivation of our notion and to explaining where the various assumptions come from. The goal is to obtain a good definition of a weak multiplier Hopf algebra. Throughout the paper, we consider the basic examples and use them, as far as possible, to illustrate what we do. In particular, we think of the finite-dimensional weak Hopf algebras. On the other hand however, we are also inspired by the far more complicated existing analytical theory.
In our forthcoming papers on the subject, we develop the theory further. In [VD-W2] we start from the definition of a weak multiplier Hopf algebra as it is obtained and motivated in this paper and we prove the main properties. In [VD-W3] we continue with the study of the source and target algebras and the corresponding source and target maps. In that paper, we also give more examples. Finally, in [VD-W4] we study integrals on weak multiplier Hopf algebras and duality. Other aspects of the theory will be considered later.