Tools for working with multiplier Hopf algebras
Tools for working with multiplier Hopf algebras
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发表时间：
2008-06
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通讯作者：
A. V. Daele
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作者：
A. V. Daele
设（A,∆）为乘数Hopf代数。一般来说，基础代数A不需要有恒等，并且副积∆不映射A到A⊗A，而是映射到它的乘数代数M （A⊗A）。在本文中，我们研究了在处理这类乘子Hopf代数时经常使用的一些工具，这些工具是在这种情况下处理无恒等代数的典型工具。其基本成分是一个一元左a模X，其基本构造是通过观察线性映射ρ: a→X满足ρ（aa′）= ρ（a′）其中a， a′∈a来扩展模。我们将模的作用写成乘法。当然，当x∈x时，当ρ(a)= ax时，我们得到这样一个线性映射。如果A有恒等矩阵，所有的线性映射都有这种形式对于x = ρ(1)然而，关键是在非一元代数的情况下，这种映射的空间通常严格大于X本身。我们得到一个扩展模块，用X表示（原因将在文中解释）。我们研究了出现这种扩展模块的各种更复杂的情况，并通过几个例子来说明这一切，从非常简单的例子到迭代扩展发挥作用的更复杂的例子。我们参考文献中出现的案例。我们使用扩展模块的基本思想，以一种更严格的方式来解释所谓的覆盖技术，这是在使用Sweedler符号来表示余积和协同时所需要的。同样，我们给出了许多例子，并参考了应用该技术的现有文献。
Let (A, ∆) be a multiplier Hopf algebra. In general, the underlying algebra A need not have an identity and the coproduct ∆ does not map A into A ⊗ A but rather into its multiplier algebra M (A ⊗ A). In this paper, we study some tools that are frequently used when dealing with such multiplier Hopf algebras and that are typical for working with algebras without identity in this context. The basic ingredient is a unital left A-module X, and the basic construction is that of extending the module by looking at linear maps ρ : A → X satisfying ρ(aa � )= aρ(a � ) where a, a � ∈ A. We write the module action as multiplication. Of course, when x ∈ X ,a nd when ρ(a )= ax, we get such a linear map. And if A has an identity, all linear maps ρ have this form for x = ρ(1). However, the point is that in the case of a non-unital algebra, the space of such maps is in general strictly bigger than X itself. We get an extended module, denoted by X (for reasons that will be explained in the paper). We study all sorts of more complicated situations where such extended modules occur and we illustrate all of this with several examples, from very simple ones to more complex ones where iterated extensions come into play. We refer to cases that appear in the literature. We use this basic idea of extending modules to explain, in a more rigorous way, the so-called covering technique, which is needed when using Sweedler’s notations for coproducts and coactions. Again, we give many examples and refer to the existing literature where this technique is applied.