Weak Multiplier Hopf Algebras II: Source and Target Algebras

Let (A,Delta) be a weak multiplier Hopf algebra. It is a pair of a non-degenerate algebra A, with or without identity, and a coproduct Delta : A -> M(A circle times A), satisfying certain properties. In this paper, we continue the study of these objects and construct new examples. A symmetric pair of the source and target maps epsilon(s) and epsilon(t) are studied, and their symmetric pair of images, the source algebra and the target algebra epsilon(s)(A) and epsilon(t)(A), are also investigated. We show that the canonical idempotent E (which is eventually Delta(1)) belongs to the multiplier algebra M(B circle times C), where (B=epsilon(s)(A), C=epsilon(t)(A)) is the symmetric pair of source algebra and target algebra, and also that E is a separability idempotent (as studied). If the weak multiplier Hopf algebra is regular, then also E is a regular separability idempotent. We also see how, for any weak multiplier Hopf algebra (A,Delta), it is possible to make C circle times B (with B and C as above) into a new weak multiplier Hopf algebra. In a sense, it forgets the 'Hopf algebra part' of the original weak multiplier Hopf algebra and only remembers symmetric pair of the source and target algebras. It is in turn generalized to the case of any symmetric pair of non-degenerate algebras B and C with a separability idempotent E is an element of M(B circle times C). We get another example using this theory associated to any discrete quantum group. Finally, we also consider the well-known 'quantization' of the groupoid that comes from an action of a group on a set. All these constructions provide interesting new examples of weak multiplier Hopf algebras (that are not weak Hopf algebras introduced).