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 Δ on A making the pair (A,Δ) 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 A1 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 Δ and define Δ1 as [Formula: see text] where f ∈ A1, p,q ∈ G and where n is the number of elements in the subgroup H. Then Δ1 will leave the subalgebra invariant (in the sense that the image is in the multiplier algebra M(A1 ⊗ A1) 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.