Bicrossproducts of multiplier Hopf algebras

Abstract

In this paper, we generalize Majid's 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 Delta(#) on A # B. We discuss in detail the necessary compatibility conditions between the action and the coaction for Delta(#) to be a proper coproduct on A # B. The result is again a regular multiplier Hopf algebra. Majid's 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 = KH and H boolean AND 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. 2011 (C) Elsevier Inc. All rights reserved.