Algebraic quantum groups II. Constructions and examples

TL;DR

This paper explores the construction of algebraic quantum hypergroups from algebraic quantum groups, generalizing previous results by removing *-structure assumptions and allowing group-like projections in the multiplier algebra, leading to new examples.

Contribution

It extends the theory of algebraic quantum hypergroups by removing *-structure constraints and considering projections in the multiplier algebra, resulting in new and known examples.

Findings

Constructs algebraic quantum hypergroups from algebraic quantum groups.

Generalizes previous results by removing *-structure assumptions.

Provides new examples of algebraic quantum hypergroups.

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 on A making it a multiplier Hopf algebra. In fact, because there exist integrals, we get an algebraic quantum group. Now let H be a finite subgroup of G and consider the subalgebra 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 it so that it will leave the subalgebra invariant (in the sense that the image is in the multiplier algebra of the tensor product of this subalgebra with itself). However, the modified coproduct on the subalgebra 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. Group-like projections in a *-algebraic quantum group A 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. In this paper we push this result further. On the one hand, we no longer assume the *-structure 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.