Tensor Categories and Representations of Quantized Algebras

表示论是对空间对称性的研究，例如我们的3维空间，或者更广泛地说，具有任意(甚至无穷多个)维度的空间。在这一理论中，对称性由该空间的线性变换来表示，或者更明确地说，由矩阵来表示。因此，给定对称结构的表示基本上是满足某一自然非线性方程组的矩阵的集合。这些方程是由我们所代表的对称结构的确切类型决定的--群、李代数或结合代数。给定结构的表示本身形成了一个相当复杂和丰富的结构，该结构对不同表示之间的关系(或映射)进行编码。这种更高级别的结构被称为表示范畴。对于某些类型的结构(例如，对于群、李代数、量子群)，表示可以相乘；在这种情况下，对应的类别是张量范畴(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是非常有趣的，而且许多张量范畴并不是作为表示范畴出现的。PI将调查普通范畴和张量范畴，其中一些作为表示范畴出现，另一些不是，以及它们之间的联系。特别地，将研究P.Deligne提出的表示范畴的复数秩泛化。粗略地说，这是允许矩阵的集合或行的元素的数目为非整数的推广。当人们感兴趣的不变量变成元素或行数的多项式时，这种看似荒谬的设置变得有意义和有用，这通常是真的。PI还将研究奇异辛簇的量化，例如辛分解。这些非对易代数出现在最近感兴趣的某些类型的量子场论中，作为量子可观察到的代数。这个项目为研究生提供了研究培训的机会。这个项目涉及的研究包括：张量范畴；量子群；复数阶表示理论；Cherednik代数；量子化的短星积；几何朗兰兹程序的解析方法。PI的工作计划如下所示。1.发展了特征p和Frobenius正则范畴中对称张量范畴的Frobenius函子理论；对融合范畴特别是扭曲Deligne积的精确分解进行了分类；在小量子群的表示范畴上对纤维函子和模范畴进行了分类；计算了小特征约化群的倾斜模范畴的半简化，并用它计算了模p的倾斜模的维数；证明了辫子融合范畴的表示的拟模性；在特征p2中构造了类似于特征2中的Etingof-Benson范畴的新的对称张量范畴；计算了这些范畴的上同调；发展了Verlinde范畴中的李理论；发展了辛反射融合范畴的理论；继续发展了有限维Hopf代数在除法代数(特别是域)上的作用理论；分类了酉张量范畴。利用动力学Weyl群研究Casimir联络下Toledano Laredo的单数定理的离散模拟，研究|q|=1的量子群表示的特征。2.继续发展P.Deligne的思想，并将各种经典结构(包含对称群S_n或经典李群GL(N)，O(N)，Sp(2n))的表示理论推广到秩参数n的复值。这些结构将包括退化的仿射Hecke代数，有理和三角Cherednik代数，辛反射代数，实还原李群(即对称对)，李超代数，仿射李代数，(抛物线)范畴O，用于约化李代数、延吉安和其他结构。计算可约轨迹，得到这些表示理论中的各种特征公式和签名公式，并回答在经典背景下已知感兴趣的各种其他表示理论问题。3.研究了双杨氏表示理论、椭圆代数理论、分圆Cherednik代数的表示、Cherednik代数表示的签名、Cherednik代数的正特征表示、Cherednik代数的正像函子和逆象函子。4.继续发展过滤量化的短星积理论。5.继续与E.Frenkel和D.Kazhdan合作，对几何朗兰兹通信进行分析。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Representation theory is a study of symmetries of space, such as our 3-dimensional space, or, more generally, a space with any (even infinite number) of dimensions. In this theory, symmetries are represented by linear transformations of this space, or, more explicitly, by matrices. Thus, a representation of a given symmetry structure is basically a collection of matrices which satisfy a certain natural system of nonlinear equations. The equations are determined by the exact type of symmetry structure we are representing - a group, a Lie algebra, or an associative algebra. 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 type of structures (e.g. for groups, Lie algebras, quantum groups), representations can be multiplied; in this case the corresponding categories are tensor categories (as 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. The PI will investigate ordinary and tensor categories, some of which arise as representation categories and some of which don't, as well as the connections between them. In particular, complex rank generalizations of representation categories proposed by P. Deligne will be investigated. Roughly speaking, this is a generalization in which the number of elements of a set or rows of a matrix is allowed to be non-integer. This seemingly nonsensical setting becomes meaningful and useful when the invariants one is interested in turn out to be polynomials of the number of elements or rows, which is often true. The PI will also investigate quantizations of singular symplectic varieties, for instance symplectic resolutions. These are non-commutative algebras that appear in certain kinds of quantum field theories of recent interest as algebras of quantum observables. This project provides research training opportunities for graduate students.This project involves research on: tensor categories; quantum groups; representation theory in complex rank; cherednik algebras; short star-products on quantizations; analytic approach to Geometric Langlands program. The plan of PI's work is as follows. 1. Develop a theory of Frobenius functors for symmetric tensor categories in characteristic p and Frobenius exact categories; classify exact factorizations of fusion categories, in particular twisted Deligne products; classify fiber functors and module categories over the representation category of the small quantum group; compute the semisimplification of the category of tilting modules for a reductive group in small characteristic, and use it to compute the dimensions of tilting modules modulo p; prove quasi-motivicity of representations of braid groups arising from braided fusion categories; construct new symmetric tensor categories in characteristic p2 similar to the Etingof-Benson categories in characteristic 2; compute cohomology of these categories; develop Lie theory in the Verlinde category; develop a theory of symplectic reflection fusion categories; continue to develop the theory of actions of finite dimensional Hopf algebras on division algebras (in particular, fields); classify unipotent tensor categories. Work on a discrete analog of the monodromy theorem of Toledano Laredo for the Casimir connection, using dynamical Weyl groups, Study signatures of representations of quantum groups for |q|=1. 2. Continue to 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 the rank parameter n. These structures will include degenerate affine Hecke algebras, rational and trigonometric Cherednik algebras, symplectic reflection algebras, real reductive Lie groups (i.e., symmetric pairs), Lie superalgebras, affine Lie algebras, (parabolic) category O for reductive Lie algebras, Yangians, and other structures. Compute reducibility loci and obtain various character formulas and signature formulas in these representation theories, and answer various other representation theoretic questions which are known to be interesting in the classical setting. 3. Work on the representation theory of double Yangians, the theory of elliptic algebras, representations of cyclotomic Cherednik algebras, signatures of representations of Cherednik algebras, representations of Cherednik algebras in positive characteristic, direct and inverse image functors for Cherednik algebras. 4. Continue to develop the theory of short star-products on filtered quantizations. 5. Continue to work with E. Frenkel and D. Kazhdan on an analytic approach to the geometric Langlands correspondence.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.