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Integrating categorical and geometric methods in non-semisimple representation theories

在非半简单表示理论中集成分类和几何方法

基本信息

批准号：
1303301
负责人：
Vera Serganova
金额：
$ 16.7万
依托单位：
University of California-Berkeley
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2013
资助国家：
美国
起止时间：
2013-09-15 至 2016-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1303301&HistoricalAwards=false
关键词：
Integrating categorical geometric methods non

项目摘要

The aim of this project is to solve several open problems in representation theory of Lie algebras and superalgebras via categorical and geometric methods. In particular, we propose to study the category T of tensor representation of sl-infinity. In this proposal this category plays two different roles. In application to infinite-dimensional Lie algebras it is the target category for categorification of boson-fermion correspondence, the Fock space equipped with action of free bosons and fermions is identified with the complexified Grothendieck group of the category T. In application to Lie superalgebras we use the approach originally suggested by Brundan. Namely, we identify the complexified Grothendieck group of the category of representation of a classical supergroup G and certain "tensor" representation M of sl-infinity, and categorify the action of the Chevalley generators of sl-infinity in M by the translation functors. Recent papers of Brundan and Stroppel, Cheng, Lam and Wang, Gruson and the author demonstrate the power of this approach by making essential progress in long standing open problems (such as character formulae, Kazhdan-Lusztig theory for the general linear superalgebras and explicit description of extension groups). We suggest to combine this approach with our results on the structure of individual sl-infinity modules in T to obtain new information about modules over Lie superalgebras: description of tensor products, calculation of superdimension, Kazhdan-Lusztig theory for the orthosymplectic superalgebras. Another goal is to generalize T to the case of fields of positive characteristic, and explore connection with Deligne's tensor categories. We also combine the above approach with geometric methods in representation theory of Lie superalgebras. In particular, we introduce several conjectures relating thick ideals of the tensor category G-mod for a classical supergroup G, equivariant sheaves on the self-commuting cone of G and the socle filtration of the corresponding sl-infinity modules. The last part of proposal addresses the problem of generalizing Borel-Weil-Bott theorem for supergroups. Although partial results in this area were obtained almost 30 years ago, a complete answer is still unknown.Supersymmetry is an important tool in modern theoretical physics. The methods of supersymmetry factor the questions interesting for physicists through the theory of representations of supergroups and superalgebras. There are still a lot of gaps in mathematical foundations in this subject: the methods familiar from representation theory of reductive groups get stuck early since the algebraic structure of representations is significantly more complicated. As new phenomena were discovered, it turned out that they have analogues in other non-semisimple branches of representation theory: modular representations and representations of infinite-dimensional Lie algebras. The latter representations have a wide range of applications: from integrable systems to string theory. We suggest a new approach to a key tool of this theory, vertex operators, by categorification of boson-fermion correspondence. We also plan to study further the duality between representation of infinite-dimensional Lie algebras and classical supergroups.

这个项目的目的是通过范畴和几何方法解决李代数和超代数表示论中的几个公开问题。特别地，我们建议研究SL-无限的张量表示范畴T。在本提案中，这一类别起着两种不同的作用。在无限维李代数中，它是玻色子-费米子对应范畴化的目标范畴，具有自由玻色子和费米子作用的Fock空间被等同于范畴T的复化Grothendieck群。在李超代数的应用中，我们使用Brundan最初提出的方法。也就是说，我们确定了经典超群G的表示范畴的复化Grothendieck群和sl-无限的某个“张量”表示M，并通过平移函子对sl-无限的Chevalley生成元在M中的作用进行了分类。最近的论文布伦丹和Stroppel，郑，林和王，格鲁森和作者证明了这种方法的力量作出重大进展，长期悬而未决的问题（如字符公式，Kazhdan-Lusztig理论的一般线性超代数和明确的描述的扩展群）。我们建议将这种方法联合收割机与我们关于T中单个sl-无穷模的结构的结果结合起来，以获得关于李超代数上模的新信息：张量积的描述，超维数的计算，正交辛超代数的Kazhdan-Lusztig理论. 另一个目标是推广T的情况下，领域的积极特征，并探讨连接德利涅的张量范畴。我们还将上述方法与李超代数表示论中的几何方法结合起来，得到了联合收割机。特别地，我们介绍了几个与经典超群G的张量范畴G-mod的厚理想、G的自对易锥上的等变层以及相应的sl-无穷模的柱脚滤子有关的定理。建议的最后一部分解决的问题，推广Borel-Weil-Bott定理的超群。虽然在这方面的部分结果已在近30年前获得，但一个完整的答案仍然是未知的。超对称的方法通过超群和超代数的表示理论来解决物理学家感兴趣的问题。在这个问题的数学基础上仍然有很多空白：从还原群的表示论中熟悉的方法很早就卡住了，因为表示的代数结构要复杂得多。随着新现象的发现，它们在表示论的其他非半单分支中也有类似的东西：模表示和无限维李代数的表示。后一种表示有着广泛的应用：从可积系统到弦理论。我们提出了一种新的方法，这个理论的一个关键工具，顶点运营商，玻色子费米子对应的分类。我们还计划进一步研究无限维李代数的表示与经典超群之间的对偶性。