ABSTRACT
For any group G, the group algebra kG over a field k has a natural Hopf alge‐bra structure. If the group is finite, this Hopf algebra is finite‐dimensional and so the dual is again a (finite‐dimensional) Hopf algebra. This is no longer the case when G is not finite. However, by taking functions with finite support on G, one has a reduced dual which is a multiplier Hopf alge‐bra. This is an algebra without identit together with a generalized coprod‐uct, counit and antipode. Multiplier Hopf algebras are therefore the natural objects generalizing Hopf algebras to the case where the underlying algebra possibly has no identity.
Multipher Hopf algebras where introduced in 1994. Since then, the theory is being developed very much along the same lines as usual Hopf algebra theory. The main difference, apart from techmcalities, lies in the fact that the natural duality of finite‐dimensional Hopf algebras now is extended to a duality of multiplier Hopf algebras with mtegrals. This a[lows to obtain kn own results in greater generality. Among those, we have the construction of the quantum double, the duality for actions and coactions, and others.
The aim of this paper is to describe the present status of this theory.
