Bicrossproducts of algebraic quantum groups

Details Bicrossproducts of algebraic quantum groups Bicrossproducts of algebraic quantum groups

2012-02-03

arxiv.org/abs/1202.0746v1

Mathematics

Rings and Algebras

Scientific paper

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 $\Delta_#$ on the smash product $A # B$ making the pair $(A # B,\Delta_#)$ into an algebraic quantum group. 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 $\hatA$ and $\hatB$. 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. It should be mentioned 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.

Affiliated with

Also associated with

No associations

Rating

Bicrossproducts of algebraic quantum groups does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Bicrossproducts of algebraic quantum groups, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Bicrossproducts of algebraic quantum groups will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-5258