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Tensor Categories and Representations of Quantized Algebras

量化代数的张量范畴和表示

基本信息

批准号：
2001318
负责人：
Pavel Etingof
金额：
$ 65万
依托单位：
Massachusetts Institute of Technology
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-09-01 至 2025-08-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2001318&HistoricalAwards=false
关键词：
Tensor Categories Representations Quantized Algebras

项目摘要

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.

表示论是一门研究空间对称性的学科，比如我们的三维空间，或者更一般地说，一个具有任意（甚至无限）维度的空间。在这个理论中，对称性由这个空间的线性变换表示，或者更明确地说，由矩阵表示。因此，给定对称结构的表示基本上是满足非线性方程的特定自然系统的矩阵的集合。这些方程由我们所表示的对称结构的确切类型决定--一个群、一个李代数或一个结合代数。给定结构的表示本身形成了一个相当复杂和丰富的结构，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（例如群、李代数、量子群），表示可以相乘;在这种情况下，对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非常有趣，而且许多张量范畴并不是作为表示范畴出现的。PI将研究普通和张量范畴，其中一些是作为表征范畴出现的，而另一些则不是，以及它们之间的联系。特别是，P. Deligne提出的表示范畴的复秩推广将被研究。粗略地说，这是一种推广，其中允许矩阵的集合或行的元素的数量为非整数。当人们感兴趣的不变量是元素或行数的多项式时，这种看似无意义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子化，例如辛解析。这些非对易代数出现在某些种类的量子场论最近的兴趣作为代数的量子可观。该项目为研究生提供了研究训练的机会。该项目涉及研究：张量范畴;量子群;复秩表示论; cherednik代数;量子化的短星积;几何朗兰兹程序的分析方法。PI的工作计划如下。1.发展特征p和Frobenius精确范畴中对称张量范畴的Frobenius函子理论;对融合范畴的精确因子分解进行分类，特别是扭曲Deligne乘积;对小量子群表示范畴上的纤维函子和模范畴进行分类;计算小特征约化群的倾斜模范畴的半化简，并用它来计算模p的倾斜模的维度;证明由辫子融合范畴产生的辫子群表示的准动机性;在特征p2中构造类似于特征2中的Etingof-Benson范畴的新对称张量范畴;计算这些范畴的上同调;发展Verlinde范畴中的Lie理论;发展辛反射融合范畴的理论;继续发展有限维Hopf代数在除法代数（特别是域）上的作用的理论;对幂单张量范畴进行分类。使用动态外尔群，研究卡西米尔连接的达诺·拉雷多单值定理的离散模拟，研究量子群表示的签名|Q| =1时。2.继续发展P. Deligne的思想，将各种经典结构（包括对称群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仿射Hecke代数、有理和三角Cherednik代数、辛反射代数、真实的约化李群（即，对称对），李超代数，仿射李代数，约化李代数的（抛物）范畴O，Yangians和其他结构。计算约化轨迹并获得这些表示论中的各种特征公式和签名公式，并回答已知在经典环境中有趣的各种其他表示论问题。3.工作的表示理论的双重杨吉亚，理论的椭圆代数，表示的分圆切雷德尼克代数，签名表示的切雷德尼克代数，表示的切雷德尼克代数的积极特征，直接和逆形象函子的切雷德尼克代数。4.继续发展滤子量子化的短星积理论。5.继续与E合作。Frenkel和D.该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。