Hopf algebroids and operads

大多数数学理论的第一步是展示所考虑对象的完整结构。例如，单独的整数集并不是人们在数论中所追求的结构，有加法和乘法运算，只有整体才能产生真正深刻的问题和应用。在这个研究项目中，我们将研究存在于某些数学对象上同调上的代数结构。回想一下，举个例子，拓扑空间存在（比如说积分）上同调。这是一个不变量，它编码了关于给定空间的基本信息，可以用来严格证明球体不能变形成环面。同样，在代数、拓扑或几何中，群、代数和大多数其他对象也存在上同调，这些不变量不仅在定义它们的区域内有广泛的应用。例如，物理学中某些场论的行为是通过与场相关的拓扑电荷来理解的，而这些只不过是理论所处时空的上同调的元素。那么，这些上同调有什么类型的结构呢？在几乎所有的例子中，它们都有一个自然的加法，在很多情况下，它们也有一个乘法，所以它们就像整数一样变成了一个环。然而，通常还有更多，即所谓的Gerstenhaber支架，这是第三种手术，以规定的方式与其他两种手术兼容。精确的公理看起来比加法和乘法更神秘，这可能会引发这样的问题：这种结构是否真的如此自然和迷人？幸运的是，有足够的结果，例如Kontsevich著名的形式化定理，证明了这种结构的相关性，以及更好地理解其性质可以产生多么深远的应用；有关进一步的细节，请参阅提案的主要部分。在这个项目中进行的具体研究将进一步阐明哪种类型的上同调理论存在这样的第三运算，以及由此产生的代数结构的性质告诉我们关于我们正在谈论的上同调的原始对象的什么。这个项目的一个重要方面将是研究问题的语言和背景。大致来说，有两种主要的方法来实现这一切，一种称为操作数，一种称为派生范畴，我们将研究在其中一种方法中已经得到的结果在多大程度上在另一种方法中有类似的结果。到目前为止，首席研究员一直在两种环境中的一种中工作，因此一个重要的目标是学习另一种语言，并刺激两个社区之间的互动和交流。与上同调对偶，有一个不变量称为同调——例如，复表示中有系数的有限群的上同调是该群在其上起平凡作用的表示的子空间，而同调是它在其上起平凡作用的（最大）商空间。在同调上，潜在的附加代数结构是上同调环的作用，或者是产生第二个同调概念的微分，称为循环同调。在特别好的情况下，上同构和同构是同构的，同构把格斯滕哈伯括号和循环微分联系在一起，这就是所谓的巴塔林-维尔科夫斯基代数。了解这种情况何时发生，以及这些代数结构的作用是什么，这些代数结构最初是在一个完全不同的背景下引入的，即量子场论，是这个项目的长期目标。

One of the first steps in most mathematical theories is to exhibit the full structure of the objects under consideration. For example, the set of integers alone is not the structure one is after in number theory, there are the operations of addition and multiplication, and only the whole package gives rise to truly deep questions and applications. In this research project, we will study such algebraic structures that are present on the cohomology of certain mathematical objects. Recall that there is for example the (say integral) cohomology of a topological space. This is an invariant that encodes essential information about a given space and can be used e.g. to prove rigorously that a sphere can not be deformed into a torus. Similarly there is the cohomology of a group, an algebra and most other objects in algebra, topology or geometry, and these invariants have found a wide range of applications not only within the area that has defined them. For example, the behaviour of certain field theories in physics is understood via topological charges associated to fields, and these are nothing but elements of the cohomology of the space-time on which the theory lives.Now, what type of structure do these cohomologies have? In pretty much every example they have a natural addition, and in many cases they also have a multiplication, so they become a ring like the integers. However, often there is more, namely a so-called Gerstenhaber bracket which is a third operation that is compatible with the two others in a prescribed manner. The precise axioms look more mysterious than that of an addition and a multiplication, and this might prompt the question whether this structure is really so natural and fascinating. Fortunately, there are enough results such as for example Kontsevich's famous formality theorem that demonstrate how relevant this structure is, and how far-reaching applications can emerge from a better understanding of its properties; see the main part of the proposal for further details.The concrete research that will be carried out in this project will further clarify for which type of cohomology theories there is such a third operation, and what the properties of the resulting algebraic structure tell us about the original object whose cohomology we are talking about.An important aspect of the project will be the language and setting in which the questions will be studied. There are roughly speaking two main approaches to all this, one called operads and one called derived categories, and we will investigate in how far results already obtained in one of them have analogues in the other. The principal investigator has been working in one of the two settings so far, thus an important objective is to learn also the other language, and to stimulate interaction and communication between the two communities.Dually to cohomology there is an invariant called homology - for instance, the cohomology of a finite group with coefficients in a complex representation is the subspace of the representation on which the group acts trivially, whereas the homology is the (largest) quotient space on which it does so. On homology, potential additional algebraic structures are an action of the cohomology ring, or a certain differential that gives rise to a second notion of homology called cyclic homology. In particularly nice cases, cohomology and homology turn out to be isomorphic, and the isomorphism relates the Gerstenhaber bracket and the cyclic differential in what is called a Batalin-Vilkovisky algebra. To understand when this happens and what is the role of these algebraic structures that first were introduced in a completely different context, namely quantum field theory, is a long-term objective of this project.