2009/03/17 by Lydia Delvaux, Delvaux, Lydia, Alfons Van Daele +3
Mathematics · #16W30 #FOS: Mathematics #Quantum Algebra (math.QA) #Rings and Algebras (math.RA) #math.QA #math.RA #msc:16W30
paper · pdf · doi:10.48550/arxiv.0903.2974
arxiv created 2009/03/17 · arxiv updated 2009/12/01
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 this 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 the intersection of H and K is trivial 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.