A Survey on Multiplier Hopf Algebras

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. We recall and discuss the two notions in the first sections of the paper. 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 ∆# on A#B. We will discuss in detail the necessary compatibility conditions between the action and the coaction for ∆# 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 will 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 will yield a duality between A#B and the smash product C#D. We show that the bicrossproduct of algebraic quantum groups is again an algebraic quantum group (i.e. a regular multiplier Hopf algebra with integrals). The -algebra case will also be considered. Some special cases will be treated and they will be related with other constructions available in the literature. Finally, the basic example, coming from a (not necessarily finite) group G with two subgroups H and K such that G = KH and H ∩K = {e} (where e is the identity of G) will be used throughout the paper for motivation and illustration of the different notions and results. The cases where either H or K is a normal subgroup will get special attention. March 2009 (Version 2.1) 1 Department of Mathematics, Hasselt University, Agoralaan, B-3590 Diepenbeek (Belgium). E-mail: Lydia.Delvaux@uhasselt.be 2 Department of Mathematics, K.U. Leuven, Celestijnenlaan 200B (bus 2400), B-3001 Heverlee (Belgium). E-mail: Alfons.VanDaele@wis.kuleuven.be 3 Department of Mathematics, Southeast University, Nanjing 210096, China. E-mail: shuanhwang@seu.edu.cn