Tensor Categories and Representation Theory

表征理论是对空间对称性的研究，比如我们的三维空间，或者更一般地说，具有任意维数（甚至无限）的空间。在这个理论中，底层空间的对称性被编码在一个代数结构中，代数结构中的元素用线性变换来表示，或者更明确地说，用矩阵来表示。因此，表示基本上是满足一定自然非线性方程组的矩阵集合。这些方程是由正在研究的对称性集合决定的。给定结构的表示本身形成了一个相当复杂和丰富的结构，它编码了不同表示之间的关系（或映射）。这种更高层次的结构被称为表征范畴。对于某些类型的结构（如群、李代数和量子群），表示可以相乘；在这种情况下，相应的范畴是张量范畴，因为表示的乘法类似于张量的乘法。事实证明张量范畴的概念本身就很有趣，很多张量范畴并不是作为表示范畴出现的。本研究涉及普通类别和张量类别，其中一些是作为表征类别出现的，而另一些则不是，并研究它们之间的联系。该项目特别研究了P. Deligne提出的表征类别的复秩概化。粗略地说，这是一个泛化，其中一个集合的元素数或一个矩阵的行数被允许是非积分的。当感兴趣的不变量变成元素数或行数的多项式时，这就变得有意义和有用了，这种情况经常发生。该项目还涉及描述量子系统隐藏对称性的量子群的研究，以及导致不变量的屈服张量类别，使我们能够区分等效结和链接。本研究项目将研究张量范畴；环上的Hopf代数作用；量子组;复秩表示理论；Hecke代数，Cherednik代数，辛反射代数；非交换代数;和泊松同源性。PI的计划如下。(1)建立有限维Hopf代数对除法代数（特别是场）的作用理论，并应用于证明Hopf作用的不存在性命题，建立张量范畴的扩展理论，对单能范畴进行分类，对小量子群的纤维函子和模范畴进行分类。(2)利用动态Weyl群研究了Casimir连接下Toledano Laredo单一性定理的离散模拟；量子仿射代数的迹函数|q|=1时量子群表示的签名。(3)发展了P. Deligne的思想，将各种经典结构(包含对称群S(n)或经典李群GL(n)， O（n), Sp(2n)）的表示理论推广到n的复值。(4)研究了双yangian的表示理论，曲线上Cherednik代数的表示理论，椭圆代数上Cherednik代数的表示理论，以及有理Cherednik代数表示的签名。(5)指导本科生和高中生关于关联代数下中心级数的作业。

Representation theory is a study of symmetries of space, such as our 3-dimensional space, or, more generally, a space with any number of dimensions (even infinite). In this theory, symmetries of the underlying space are encoded in an algebraic structure and the elements in the algebraic structure are represented by linear transformations, or, more explicitly, by matrices. Thus, a representation is basically a collection of matrices that satisfy a certain natural system of nonlinear equations. These equations are determined by the collection of symmetries that are being studied. Representations of a given structure themselves form a quite intricate and rich structure, which encodes relations (or mappings) between different representations. This higher-level structure is called the category of representations. For some types of structures (e.g., for groups, Lie algebras, and quantum groups), representations can be multiplied; in this case the corresponding categories are tensor categories because multiplication of representations is similar to multiplication of tensors. It turns out that the notion of a tensor category is very interesting in its own right, and that many tensor categories don't arise as categories of representations. This research concerns ordinary and tensor categories, some of which arise as representation categories and some of which don't, and to study connections between them. In particular, the project studies complex rank generalizations of representation categories proposed by P. Deligne. Roughly speaking, this is a generalization in which the number of elements of a set or rows of a matrix is allowed to be nonintegral. This becomes meaningful and useful when the invariants of interest turn out to be polynomials of the number of elements or rows, which is often the case. The project also involves the study of quantum groups which describe hidden symmetries of quantum systems, and yield tensor categories which lead to invariants allowing us to distinguish in-equivalent knots and links. This research project will study tensor categories; Hopf algebra actions on rings; quantum groups; representation theory in complex rank; Hecke algebras, Cherednik algebras, symplectic reflection algebras; noncommutative algebra; and Poisson homology. The PI's plan is as follows. (1) Develop a theory of actions of finite dimensional Hopf algebras on division algebras (in particular, fields) and apply it to proving non-existence statements for Hopf actions, develop a theory of extensions of tensor categories, classify unipotent categories, and classify fiber functors and module categories for the small quantum group. (2) Study a discrete analog of the monodromy theorem of Toledano Laredo for the Casimir connection, using dynamical Weyl groups; trace functions for quantum affine algebras; and signatures of representations of quantum groups for |q|=1. (3) Develop the ideas of P. Deligne, and extend representation theories of various classical structures (containing the symmetric group S(n) or classical Lie groups GL(n), O(n), Sp(2n)) to complex values of n. (4) Study the representation theory of double Yangians, of Cherednik algebras on curves, on elliptic algebras, and on signatures of representations of rational Cherednik algebras. (5) Supervise the work of undergraduate and high school students on the lower central series of associative algebras.