Bicrossproducts of multiplier Hopf algebras

Abstract

In this paper, we generalize Majidʼs bicrossproduct construction. We start with a pair ( A , B ) of two regular multiplier Hopf algebras. We assume that B is a right A-module algebra and that A is a left B-comodule coalgebra. The right action of A on B gives rise to the smash product A # B . The left coaction of B on A gives a possible coproduct Δ # on A # B . We discuss in detail the necessary compatibility conditions between the action and the coaction for Δ # to be a proper coproduct on A # B . The result is again a regular multiplier Hopf algebra. Majidʼs construction is obtained when we have Hopf algebras. We also look at the dual case, constructed from a pair ( C , D ) of regular multiplier Hopf algebras where now C is a left D-module algebra while D is a right C-comodule coalgebra. We show that indeed, these two constructions are dual to each other in the sense that a natural pairing of A with C and of B with D yields a duality between A # B and the smash product C # D . We show that the bicrossproduct of an algebraic quantum group is again an algebraic quantum group (i.e. a regular multiplier Hopf algebra with integrals). The ⁎-algebra case is also considered. Some special cases are treated and they are related with other constructions available in the literature. The basic example, coming from a (not necessarily finite) group G with two subgroups H and K such that G = K H and H ∩ K = { e } (where e is the identity of G) is used to illustrate our theory. More examples will be considered in forthcoming papers on the subject.