For any group G, the algebra K(G) of complex functions on G (with pointwise operations) is a multiplier Hopf algebra when the coproduct Δ on K(G) is defined as usual by Δ(f)(p,q)=f(pq) when p,q∈G. If G is not a group, but only a groupoid, it is still possible to define a coproduct as above, provided we let Δ(f)(p,q)=0 when pq is not defined. Then (K(G),Δ) is a weak multiplier Hopf algebra. Recall that the difference between a Hopf algebra and a weak Hopf algebra lies in the fact that the coproduct is no longer assumed to be unital. The difference between a multiplier Hopf algebra and a weak multiplier Hopf algebra is similar.

Weak multiplier Hopf algebras can be considered as quantum groupoids. However, in some sense, the theory is too restrictive. A more adequate notion is that of a multiplier Hopf algebroid. Roughly speaking, in the case of a multiplier Hopf algebroid, it is no longer assumed that the base algebra is separable (in the sense of ring theory). Recall that the base algebra in the case of a groupoid as above, is the algebra of complex functions with finite support on the set of units of the groupoid. In this case, the base algebra is always separable, but this need not be so for general quantum groupoids.

In this talk, I plan to discuss the relation of these two notions, weak multiplier Hopf algebras and multiplier Hopf algebroids. Of course, I will first give precise definitions of the two concepts. A simple example will be given to explain the difference. At the end of the talk I will say something about the importance of this difference for the more involved theory of locally compact quantum groupoids and the relation with the work on measured quantum groupoid (as studied by Enock, Lesieur and others).

This is about work in progress with Shuanhong Wang (Southeast University of Nanjing - China), Thomas Timmermann (University of Mu¨nster - Germany) and Byung-Jay Kahng (Canisius College Buffalo - USA).