Tensor categories, quantum groups, and Hecke algebras

本课题提出张量范畴的研究；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;泊松同源性。PI的工作计划如下1)继续发展有限张量范畴的一般理论，特别是融合范畴的一般理论。2)利用动态Weyl群证明了关于Casimir连接的Toledano Laredo单一性定理的离散类比，证明了关于迹函数的Felder-Varchenko猜想，并研究了量子实移代数的三角、离散和离散三角类比。3)研究了有理Cherednik代数和辛反射代数的有限维表示、连续Hecke代数的表示、Cherednik代数的酉表示。研究将这些代数的表示论与李论和量子群的表示论联系起来的各种构造。4)发展P. Deligne的思想，将各种经典结构（包含对称群或经典李群）的表示理论推广到秩参数n的复值（这些结构包括简并仿射Hecke代数、有理和三角Cherednik代数、辛反射代数、实约李群、李超代数、仿射李代数、约李代数的抛物范畴O、yangian和其他结构）。5)研究了乘法颤变的量化，以及结合代数的下中心级数的结构。6)继续研究泊松变种的第零泊松同源性的结构。表征理论是对向量空间对称性的研究。在这个理论中，对称是由这个空间的线性变换（矩阵）来表示的。因此，一个给定对称结构的表示基本上是满足一定的非线性关系自然系统的矩阵的集合。这些关系由要表示的结构的确切类型决定，例如群、李代数或关联代数。更高层次的结构称为表示范畴。对于某些类型的结构（如群、李代数、量子群），表示可以相乘形成张量范畴。本项目拟研究许多普通和张量范畴，其中一些是作为表示范畴出现的，而另一些则不是，并研究它们之间的联系。PI建议研究Deligne提出的表示范畴的复秩推广。粗略地说，这是一个泛化，其中矩阵的行数被允许为非整数。当有趣的不变量是行数的多项式时，这个看似无意义的设置变得有意义和有用。

This project proposes research on tensor categories; quantum groups; representation theory in complex rank; Hecke algebras, Cherednik algebras, symplectic reflection algebras; noncommutative algebra; Poisson homology. The PI's work plan is as follows. 1) Continue developing the general theory of finite tensor categories, in particular, of fusion categories. 2) Prove a discrete analog of the monodromy theorem of Toledano Laredo for the Casimir connection, using dynamical Weyl groups, prove the Felder-Varchenko conjectures on trace functions, and study the trigonometric, the discrete, and the discrete trigonometric analogs of the quantum shift-of-the-argument algebra. 3) Study finite dimensional representations of rational Cherednik algebras and symplectic reflection algebras, representations of continuous Hecke algebras, unitary representations of Cherednik algebras. Study various constructions which link representation theory of these algebras with Lie theory and the representation theory of quantum groups. 4) Develop the ideas of P. Deligne, and extend representation theories of various classical structures (containing the symmetric group or classical Lie groups) to complex values of the rank parameter n (these structures include degenerate affine Hecke algebras, rational and trigonometric Cherednik algebras, symplectic reflection algebras, real reductive Lie groups, Lie superalgebras, affine Lie algebras, parabolic category O for reductive Lie algebras, Yangians, and other structures). 5) Work on quantizations of multiplicative quiver varieties, and on the structure of the lower central series of associative algebras. 6) Continue to study the structure of the zeroth Poisson homology of Poisson varieties.Representation theory is a study of symmetry in a vector space. In this theory, symmetries are represented by linear transformations of this space (by matrices). Thus, a representation of a given symmetry structure is basically a collection of matrices which satisfy a certain natural system of nonlinear relations. The relations are determined by the exact type of structure to be represented such as a group, a Lie algebra, or an associative algebra. A higher-level structure is called the category of representations. For some type of structures (e.g. for groups, Lie algebras, quantum groups), representations can be multiplied to form tensor categories. The present project proposes to study many ordinary and tensor categories, some of which arise as representation categories and some of which don't, and to study connections between them. The PI proposes to study complex rank generalizations of representation categories proposed by Deligne. Roughly speaking, this is a generalization in which the number of rows of a matrix is allowed to be non-integer. This seemingly nonsensical setting becomes meaningful and useful in a situation when the interesting invariants are polynomials of the number of rows.