Bicrossproducts of Algebraic Quantum Groups

Abstract

Let A and B be two algebraic quantum groups (i.e. multiplier Hopf algebras with integrals). Assume that B is a right A-module algebra and that A is a left B-comodule coalgebra. If the action and coaction are matched, it is possible to define a coproduct ∆ # on the smash product A#B making the pair (A#B, ∆ # ) into an algebraic quantum group. This result is proven in 'Bicrossproducts of multiplier Hopf algebras' (reference [De-VD-W]) where the precise matching conditions are explained in detail, as well as the construction of this bicrossproduct. In this paper, we continue the study of these objects. First, we study the various data of the bicrossproduct A#B, such as the modular automorphisms, the modular elements, ... and obtain formulas in terms of the data of the components A and B. Secondly, we look at the dual of A#B (in the sense of algebraic quantum groups) and we show it is itself a bicrossproduct (of the second type) of the duals A and B. The result is immediate for finite-dimensional Hopf algebras and therefore it is expected also for algebraic quantum groups. However, it turns out that some aspects involve a careful argument, mainly due to the fact that coproducts and coactions have ranges in the multiplier algebras of the tensor products and not in the tensor product itself. Finally, we also treat some examples in this paper. We have included some of the examples that are known for finite-dimensional Hopf algebras and show how they can also be obtained for more general algebraic quantum groups. We also give some examples that are more typical for algebraic quantum groups. In particular, we focus on the extra structure, provided by the integrals and associated objects. In [L-VD] related examples are considered, involving totally disconnected locally compact groups, but as these examples require some analysis, they fall outside the scope of this paper. It should be mentioned however that with examples of bicrossproducts of algebraic quantum groups, we do get examples that are essentially different from those commonly known in Hopf algebra theory.February 2012 (Version 1.0) Actions and coactions, integrals and cointegralsConsider two regular multiplier Hopf algebras A and B. Let ⊳ be a right action of A on B making B into a right A-module algebra. Let Γ be a left coaction of B on A making A a left B-comodule coalgebra. Assume that the action and coaction make of (A, B) a matched pair as discussed already in the introduction.In this section, we will consider integrals and cointegrals for the components A and B and we will investigate the relation with the action and the coaction. The case where B has cointegralsWe will not be able to deduce very much however, but fortunately, this case is not so important. We mainly include it for motivational purposes. In particular, we will use this case to explain what kind of problems we encounter.Recall that an element h ∈ B is called a left cointegral if it is non-zero and if bh = ε(b)h for all b ∈ B. Similarly, a non-zero element k ∈ B satisfying kb = ε(b)k for all ...