代数量子超群
Algebraic Quantum Hypergroups
摘要：
代数量子群是一个具有积分的乘子霍普夫代数。在本文中，我们将发展一个理论的代数量子超群。它与代数量子群的理论非常相似，只是不再假设余乘是同态。我们仍然需要存在一个左积分和一个右积分。也有一个对极，但它的特征在于这些积分。我们构造的对偶，就像在代数量子群的情况下，我们表明，对偶的对偶是原来的量子超群。我们定义了紧型和离散型代数量子超群，并证明了这两种类型是对偶的。紧型代数量子超群本质上是紧量子超群的代数成分，由Chapovsky和Vainerman引入并研究（在算子代数背景下）。我们将给出一些基本的例子来说明该理论的不同方面。在单独的注释中，我们将考虑更多的特殊情况和更复杂的例子。特别是，在那篇笔记中，我们将给出一个一般的构造过程，并展示这些代数量子超群的已知例子如何适合这个框架。
英文摘要：
An algebraic quantum group is a multiplier Hopf algebra with integrals. In this paper we will develop a theory of algebraic quantum hypergroups. It is very similar to the theory of algebraic quantum groups, except that the comultiplication is no longer assumed to be a homomorphism. We still require the existence of a left and of a right integral. There is also an antipode but it is characterized in terms of these integrals. We construct the dual, just as in the case of algebraic quantum groups and we show that the dual of the dual is the original quantum hypergroup. We define algebraic quantum hypergroups of compact type and discrete type and we show that these types are dual to each other. The algebraic quantum hypergroups of compact type are essentially the algebraic ingredients of the compact quantum hypergroups as introduced and studied (in an operator algebraic context) by Chapovsky and Vainerman. We will give some basic examples in order to illustrate different aspects of the theory. In a separate note, we will consider more special cases and more complicated examples. In particular, in that note, we will give a general construction procedure and show how known examples of these algebraic quantum hypergroups fit into this framework.
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