Configurations of spherical twists and derived Picard groups of Brauer graph algebras

近年来，辛几何与表示论之间的相互作用取得了丰硕的成果。其中，它在理解派生范畴的结构、等价分类和柔和代数的自等价群方面取得了重大进展，为表示论提供了一个重要的可管理的测试类。本项目的目的是在Brauer图代数及其分次类似物的研究中进一步开发和扩展这种联系。Brauer图代数是模表示理论中的自然产物。粗略地说，一个布劳尔图代数B_Γ可以被分配给(最小地)嵌入到一个曲面Γ中的任何图Σ_Γ和一个分配给图的每个顶点的自然数，称为重数。Brauer图代数与辛几何之间的联系是我与S.Opper共同研究的第一个问题。在这里，我们引入了一类包含Brauer图代数的A-无穷范畴。这些A-无穷范畴与部分包络Fukaya曲面范畴的平凡扩张是准等价的，这些曲面的边界上有一个线场和有限个停止点。研究这类A-无穷范畴得到了普通Brauer图代数的一个完全导出等价分类.由于具有平凡重数的Brauer图代数上的投射模提供了球面对象的构形，因此得到了Brauer图代数及其分次类似物的有界导出范畴的Picard群或自等价群.增强三角化类别中的球形对象产生这些类别的自等价，称为球面扭曲。它们是由Seidel和Thomas引入的，其动机源于同调镜像对称猜想，作为与拉格朗日球面相关的广义Dehn扭曲的对应物。这个项目的目的是研究Brauer图代数的有界导出范畴及其分次类似物的自等价群。一般来说，这些群体很难研究，而且这是在极少数情况下完成的。作为项目的一部分，我计划1)研究分次Brauer图代数B_Γ的射影模上球面扭曲生成的自等价群的子群，并证明这个子群同构于曲面Σ_Γ的辫子扭曲群；2)研究分次Brauer图代数的内在形式性，以便将(1)中的结果转移到任何具有适当球面对象配置的增强三角范畴(猜想，这样的范畴出现在某些Fukaya范畴和簇范畴的研究中)；3)利用Σ_Γ、移位和B_Γ的外自同构群的映射类群的某些子群，得到了布劳尔图代数的有界导范畴的自等价群的完整刻画。

The interplay between symplectic geometry and representation theory has been very fruitful in recent years. Among others, it led to a significant progress in understanding the structure of derived categories, their equivalence classification and groups of autoequivalences for gentle algebras - a class of algebras, providing an important manageable test class in representation theory. This project aims at further exploiting and extending this connection in the study of Brauer graph algebras and their graded analogues. Brauer graph algebras arise naturally in modular representation theory. Roughly speaking, a Brauer graph algebra B_Γ can be assigned to any graph Γ (minimally) embedded into a surface Σ_Γ and a natural number assigned to each vertex of the graph, called multiplicity. The connection between Brauer graph algebras and symplectic geometry was first investigated in my joint work with S. Opper. There we introduced a class of A-infinity categories, containing Brauer graph algebras. These A-infinity categories are quasi-equivalent to trivial extensions of partially wrapped Fukaya categories of surfaces with boundary equipped with a line field and a finite number of stops on the boundary. Studying this class of A-infinity categories led to a complete derived equivalence classification of ordinary Brauer graph algebras.Derived Picard groups or groups of autoequivalences of bounded derived categories of Brauer graph algebras and their graded analogues are of particular interest, since projective modules over Brauer graph algebras with trivial multiplicities provide configurations of spherical objects. Spherical objects in enhanced triangulated categories produce autoequivalences of these categories called spherical twists. They were introduced by Seidel and Thomas with motivation stemming from the homological mirror symmetry conjecture as counterparts of generalized Dehn twists associated with Lagrangian spheres. The aim of this project is to study groups of autoequivalences of the bounded derived category of Brauer graph algebras and their graded analogues. These groups are very hard to study in general and this was done in a very few cases. As part of the project I am planning to 1) study the subgroup of the group of autoequivalences generated by spherical twists along projective modules of a graded Brauer graph algebra B_Γ and show that this subgroup is isomorphic to the braid twist group of the surface Σ_Γ; 2) study intrinsic formality of graded Brauer graph algebras in order to transfer results obtained in (1) to any enhanced triangulated category with a suitable configuration of spherical objects (conjecturally, such categories appear as certain Fukaya categories and in the study of cluster categories); 3) obtain a complete description of groups of autoequivalences of the bounded derived category of Brauer graph algebras using certain subgroups of the mapping class group of Σ_Γ, shift and outer automorphisms of B_Γ.