ALGEBRAIC QUANTUM HYPERGROUPS II. CONSTRUCTIONS AND EXAMPLES

Abstract

Let G be a group and let A be the algebra of complex functions on G with finite support. The product in G gives rise to a coproduct Delta on A making the pair (A, Delta) a multiplier Hopf algebra. In fact, because there exist integrals, we get an algebraic quantum group as introduced and studied in [A. Van Daele, Adv. Math. 140 (1998) 323]. Now let H be a finite subgroup of G and consider the subalgebra A(1) of functions in A that are constant on double cosets of H. The coproduct in general will not leave this algebra invariant but we can modify Delta and define Delta(1) as Delta(1)(f)(p, q) = 1/n Sigma(r is an element of H) f(prq) where f is an element of A(1), p, q is an element of G and where n is the number of elements in the subgroup H. Then Delta(1) will leave the subalgebra invariant (in the sense that the image is in the multiplier algebra M(A(1) circle times A(1)) of the tensor product). However, it will no longer be an algebra map. So, in general we do not have an algebraic quantum group but a so-called algebraic quantum hypergroup as introduced and studied in [L. Delvaux and A. Van Daele, Adv. Math. 226 (2011) 1134-1167]. Group-like projections in a *-algebraic quantum group A (as defined and studied in [M. B. Landstad and A. Van Daele, arXiv: math. OA/0702458v1]) give rise, in a natural way, to *-algebraic quantum hypergroups, very much like subgroups do as above for a *-algebraic quantum group associated to a group (again see [M. B. Landstad and A. Van Daele, arXiv: math. OA/0702458v1]). In this paper we push these results further. On the one hand, we no longer assume the *-structure as in [M. B. Landstad and A. Van Daele, arXiv: math. OA/0702458v1] while on the other hand, we allow the group-like projection to belong to the multiplier algebra M(A) of A and not only to A itself. Doing so, we not only get some well-known earlier examples of algebraic quantum hypergroups but also some interesting new ones.