Integrating categorical and geometric methods in non-semisimple representation theories

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理论中的几个公开问题。特别地，我们建议研究sl-无穷大的张量表示范畴T。在本提案中，这一类别扮演着两个不同的角色。在无限维李代数中，它是玻色子-费米子对应分类的目标范畴，具有自由玻色子和费米子作用的Fock空间被证明为范畴T的复化的Grothendieck群。在应用于李超代数时，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-无穷大的某些张量表示M，并用平移函子分类了sl-无穷大的Chvalley生成元在M中的作用。Brundan和Stroppel，Cheng，Lam和Wang，Gruson以及作者最近的论文通过在长期存在的公开问题(如特征标公式，一般线性超代数的Kazhdan-Lusztig理论和扩张群的显式刻画)上取得实质性进展，证明了这种方法的有效性。我们建议将这一方法与我们关于T中单个sl-无穷大模的结构的结果相结合，以获得关于李超代数上的模的新信息：张量积的描述、超维的计算、直交辛超代数的Kazhdan-Lusztig理论。另一个目的是将T推广到正特征域的情况，并探索与Deligne张量范畴的联系。我们还将上述方法与李超代数表示理论中的几何方法相结合。特别地，我们引入了几个猜想，这些猜想与经典超群G的张量范畴G-mod的厚理想、G的自交换锥上的等变层以及相应的sl-无穷大模的基滤子有关。提案的最后一部分讨论了推广超群的Borel-Weil-Bott定理的问题。超对称性是现代理论物理中的一个重要工具，虽然在近30年前已经取得了部分结果，但仍然没有一个完整的答案。超对称的方法通过超群和超代数的表示理论来分解物理学家感兴趣的问题。这门学科在数学基础上仍然有很多空白：由于表示的代数结构明显更加复杂，从约化群的表示理论熟悉的方法很早就被卡住了。随着新现象的发现，它们在表示理论的其他非半简单分支中也有相似之处：模表示和无限维李代数的表示。后一种表示有广泛的应用：从可积系统到弦理论。通过对玻色子-费米子对应的分类，我们提出了一种新的方法来研究这一理论的关键工具--顶点算符。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。

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.