Tensor Categories and Geometric Representation Theory

科学和艺术中的一个基本思想是对称的思想。例如，人们期望技术问题的最佳解决方案是对称的（一个典型的例子是轮子——高度对称的形状，在我们的日常生活中非常有用）。在更深的层次上，人们期望所有的物理定律都应该是对称的，而这似乎是许多现代物理理论的指导原则（从爱因斯坦的相对论开始）。用数学术语来表达对称的概念是一个相当重要的问题；但是我们在这个方向上取得了很大的进步。实际上在这个问题的解中至少有三个层次的抽象：群，霍普夫代数，张量范畴。这个项目的第一部分与这个列表中最高级别的抽象有关，也就是张量类别。尽管这种高度抽象的理论在应用中是有用的：在物理学中，张量范畴的语言用于相变和表面现象的理论研究（如果我们想生产更高效的计算机，这一点很重要）；另一方面，张量范畴在计算机科学中被用作构建“量子计算机”的工作工具，量子计算机是计算机体系结构的一种新概念方法。因此，人们期望对这些分类的研究对现代技术的发展是有用的。因此，建议对张量范畴进行研究，重点是建立新的有趣的例子和解决分类问题。研究的第二部分是表征理论。我们可以把表征理论看作是研究如何在数学方程中实现对称性。因此，对于给定的对称性（由群、霍普夫代数等描述），表示理论旨在描述这种对称性在数学模型中可能出现的所有方式。在提案的第二部分中，建议使用张量范畴理论的见解来研究群的表示理论的某些问题（因此我们希望使用对称性来研究对称性，这确实是该理论的指导原则之一）。更技术性的是，在项目的第一部分，建议通过研究由有限群分级的融合类别和通过构建模块类别的新例子来寻找融合（=半简单张量）类别的新例子（然后对偶类别将提供张量类别的新例子）。第二部分是几何表示理论。表征理论中极为重要的一类感兴趣的对象是由G.~Lusztig引入并分类的特征束构成的。

One of the basic ideas in the science and art is the idea of {\em symmetry}. For exampleone expects that the best solutions of technical problems are symmetric (an archetypicalexample here is the wheel -- highly symmetric shape that is so useful in our everydaylife). On a deeper level it is expected that all the physical laws should be symmetric withrespect to a change of point of view and this appears to be the guiding principle ofmany modern physical theories (starting with Einstein's Relativity Theory). It is quite anontrivial problem to express the idea of symmetry in mathematical terms; but wehave a great progress in this direction. Actually there are at least three levels of abstractionin the solution of this problem: groups, Hopf algebras, tensor categories. The first partof this project is related with highest level of abstraction in this list, that is with tensorcategories. Despite of this high level of abstraction the theory is useful in the applications:in physics the language of tensor categories is used in theoretical study of phasetransitions and surface phenomena (and this is important if we want to produce moreefficient computers); on the other hand the tensor categories are used in computerscience as a working tool for the constructing of the "quantum computer", a newconceptual approach to the computer architecture. So one expects that the study oftensor categories would be useful in the development of modern technologies.Thus it is suggested to study the tensor categoriesfocusing on the construction of new interesting examplesand the solution of classification problems.The second part of the project is devoted to the representation theory. One can thinkabout representation theory as about the study how a symmetry can be realized inmathematical equations. Thus for a given symmetry (which is described by group,Hopf algebra etc) representation theory aims to describe all the ways this symmetrycan show up in mathematical models. In the second part of the proposal it is suggested tostudy certain questions of representation theory of groups using insightsfrom the theory of tensor categories (thus we would like to employ symmetryfor studying symmetry and this is indeed one of the guiding principles of the theory).More technically, in the first part of the projectit is suggested to find new examplesof fusion (= semisimple tensor) categories by studying fusion categories graded byfinite groups and by constructing new examples of module categories (then dualcategories will provide new examples of tensor categories). The second partdeals with geometric representation theory. An extremely important class of objects ofinterest for the representation theory is formed by character sheaves that wereintroduced and classified by G.~Lusztig.