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Hopf algebroids and operads

Hopf 代数数和运算数

基本信息

批准号：
EP/J012718/1
负责人：
Ulrich Kraehmer
金额：
$ 12.62万
依托单位：
University of Glasgow
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2012
资助国家：
英国
起止时间：
2012 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FJ012718%2F1
关键词：
Hopf algebroids operads

项目摘要

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.

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，整数集合本身并不是数论中所追求的结构，还有加法和乘法的运算，只有整个集合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的上同调上的代数结构。回想一下，例如，存在拓扑空间的（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，并且可以用来严格证明球体不能变形为环面。同样有上同调的一组，一个代数和大多数其他对象在代数，拓扑或几何，这些不变量已经发现了广泛的应用范围不仅在该地区已定义他们。例如，物理学中某些场论的行为是通过与场相关的拓扑荷来理解的，而这些拓扑荷只不过是场论赖以生存的时空的上同调的元素，那么，这些上同调有什么样的结构呢？在几乎所有的例子中，它们都有一个自然的加法，在许多情况下，它们也有一个乘法，所以它们变成了一个像整数一样的环。然而，通常还有更多，即所谓的Gerstenhaber括号，这是第三种操作，以规定的方式与其他两种操作兼容。精确的公理看起来比加法和乘法更神秘，这可能会引发这样一个问题：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著名的形式定理，证明了这种结构是多么相关，以及如何从更好地理解其属性中产生深远的应用;详见提案主体部分，本项目将开展的具体研究将进一步明确哪种类型的上同调理论存在这样的第三种运算，以及由此产生的代数结构的性质告诉我们关于我们正在讨论的上同调的原始对象的信息。该项目的一个重要方面将是研究问题的语言和环境。粗略地说，这一切有两种主要的方法，一种叫做操作数，一种叫做派生范畴，我们将研究在其中一种方法中已经得到的结果在另一种方法中有多少类似物。到目前为止，首席研究员一直在两种环境中的一种环境中工作，因此一个重要的目标是学习另一种语言，并促进两个社区之间的互动和交流。与上同调相对应的是一个称为同调的不变量-例如，系数为复表示的有限群的上同调是该群平凡作用于其上的表示的子空间，而同调是其上的（最大）商空间。在同调上，潜在的额外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，上同调和同调是同构的，同构将Gerstenhaber括号和循环微分联系在一起，称为Batalin-Vilkovisky代数。理解这种情况何时发生以及这些代数结构的作用是什么，这些代数结构首先是在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。