# Bicrossproducts of multiplier Hopf algebras - Mathematics > Rings and Algebras

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Abstract: In this paper, we generalize Majid-s bicrossproduct construction. We startwith a pair A,B of two regular multiplier Hopf algebras. We assume that B isa right A-module algebra and that A is a left B-comodule coalgebra. We recalland discuss the two notions in the first sections of the paper. The rightaction of A on B gives rise to the smash product A # B. The left coaction of Bon A gives a possible coproduct on A # B. We will discuss in detail thenecessary compatibility conditions between the action and the coaction for thisto be a proper coproduct on A # B. The result is again a regular multiplierHopf 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 regularmultiplier Hopf algebras where now C is a left D-module algebra while D is aright C-comodule coalgebra. We will show that indeed, these two constructionsare dual to each other in the sense that a natural pairing of A with C and of Bwith 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 analgebraic quantum group i.e. a regular multiplier Hopf algebra withintegrals. The *-algebra case will also be considered. Some special cases willbe treated and they will be related with other constructions available in theliterature.Finally, the basic example, coming from a not necessarily finite group Gwith two subgroups H and K such that G=KH and the intersection of H and K istrivial will be used throughout the paper for motivation and illustration ofthe different notions and results. The cases where either H or K is a normalsubgroup will get special attention.

Author: ** Lydia Delvaux, Alfons Van Daele, Shuanhong Wang**

Source: https://arxiv.org/