代数量子群的双交叉积
Bicrossproducts of algebraic quantum groups
摘要：
设$A$和$B$是两个代数量子群（即带积分的乘子Hopf代数）.设$B$是右$A$ -模代数，$A$是左$B$ -余模代数.如果作用和余作用是匹配的，则可以在smash积$A # B$上定义一个余积$Δ_#$，使$(A # B,Δ_#)$对成为一个代数量子群。在本文中，我们继续这些对象的研究。首先，我们研究了双交叉积$A # B$的各种数据，如模自同构、模元素、模映射等。并根据组分$A$和$B$的数据得到公式。其次，我们看看对偶的$A # B$（在意义上的代数量子群），我们表明它本身是一个双交叉产品（第二类）的双$\hatA$和$\hatB$。这个结果对于有限维Hopf代数是直接的，因此对于代数量子群也是可以预期的。然而，事实证明，某些方面涉及到一个仔细的论点，主要是由于这样一个事实，即余积和余作用在张量积的乘子代数中有范围，而不是在张量积本身。最后，我们还处理了一些例子在本文中。我们已经包括了一些已知的有限维Hopf代数的例子，并展示了它们如何也可以得到更一般的代数量子群。我们还给出了一些代数量子群的典型例子。特别是，我们专注于额外的结构，所提供的积分和相关的对象。应该提到的是，与代数量子群的bicrossproducts的例子，我们得到的例子是本质上不同于那些通常已知的霍普夫代数理论。
英文摘要：
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 $Δ_#$ on the smash product $A # B$ making the pair $(A # B,Δ_#)$ 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.
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