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. 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.

Galois objects for algebraic quantum groups

“…Indeed, when doing computations with Sweedler notation, it is crucial that all expressions are covered. We refer to [9] for a careful analysis of this technique, and to Appendix A for the more intuitive approach.…”

“…Although this seems simple, the situation can become quite complicated when multiple coverings are needed (see e.g. the examples in [5]). However, in our paper the situation is not so bad, probably because we are working with algebraic quantum groups in stead of general multiplier Hopf algebras: mostly it is seen at first sight if an expression is well-covered or not.…”

Classifying complements for Hopf algebras and Lie algebras

“…The aim of this paper is to give the answer to the (CCP) if C is the category of Hopf (respectively Lie) algebras. There exists a general principle: H is an A-complement of E in a given category C if and only if E ∼ = A ⊲⊳ H, where A ⊲⊳ H is a 'bicrossed product' in the category C associated to a 'matched pair ' between the objects A and H. This principle becomes a theorem when C is the category of groups or groupoids [4], algebras [5], Hopf algebras [15], Lie groups or Lie algebras [13], locally compact quantum groups [19], multiplier Hopf algebras [6]. Let H be a given A-complement of E. Hence, there exists a canonical isomorphism A ⊲⊳ H ∼ = E in C. Now, the description and the classification part of the (CCP) is obtained from the following subsequent question: describe and classify all objects H in C such that there exists an isomorphism A ⊲⊳ H ∼ = A ⊲⊳ H in C. This can also be viewed as a descent type question: the classification of all A-complements of E needs a parallel theory similar to what is called the classification of forms in the classical descent theory [12], [18].…”

Weak Multiplier Hopf Algebras II: Source and Target Algebras

“…If the algebra B is trivial, we obtain the smash product C#Q, constructed with the right action of Q on C while if C is trivial, we get the smash product Q#B, for the left action of B on C. Recall that, in the original paper [38], we developed the theory for left actions. The reader can also have a look at Section 2 of the expanded version of [40] found on arXiv where the two types of smash products are reviewed.…”