A multiplier Hopf-algebra (introduced by Van Daele) is a pair $(A, Delta)$ where $A$ is a non-degenerate algebra $A$ together with a non-degenerate algebra morphism $Delta: A to M(A otimes A)$ such that

(i) the following subsets of $M(A otimes A)$ are contained in $A otimes A$:

$$Delta(A)(1 otimes A), quad Delta(A)(A otimes 1), quad (A otimes 1) Delta(A), quad (1 otimes A)Delta(A).$$

(ii) $Delta$ is coassociative.

(iii) The following maps $A otimes A to A otimes A$ are bijective:

$$a otimes b mapsto(a otimes 1)Delta(b), quad a otimes b mapsto Delta(a)(1 otimes b)$$

Examples include:

- Any Hopf-algebra.
- The finitely supported functions on a group $X$. Here, the comultiplication is given by

$$Delta(f)(x,y) = f(xy)$$

Are there any easy other examples?