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

近年来，辛几何和表示论之间的相互作用已经非常富有成果。除其他外，它导致了一个重大的进展，在理解结构的派生类别，其等价分类和集团的自等价温柔代数-一类代数，提供了一个重要的可管理的测试类表示论。该项目旨在进一步利用和扩展这种联系在布劳尔图代数及其分次类似物的研究。Brauer图代数在模表示理论中自然出现。粗略地说，一个Brauer图代数B_Γ可以被赋给任何嵌入到一个曲面B_Γ中的图Γ，并且给图的每个顶点赋一个自然数，称为重数。Brauer图代数和辛几何之间的联系首先是在我与S。奥珀在那里，我们引入了一类A-无穷范畴，包含Brauer图代数。这些A-无穷范畴是拟等价于曲面的部分包裹福谷范畴的平凡扩张，其边界上具有一个线域和有限个停点。对这类A-无穷范畴的研究导致了普通Brauer图代数的一个完整的导出等价分类。Brauer图代数的有界导出范畴的导出Picard群或自等价群及其分次类似物特别令人感兴趣，因为Brauer图代数上具有平凡重数的投射模提供了球形对象的构型。增强三角形化类别中的球形对象产生这些类别的自等价，称为球形扭曲。它们是由赛德尔和托马斯引入的，其动机来自同调镜像对称猜想，作为与拉格朗日球相关的广义德恩扭曲的对应物。 本项目的目的是研究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，这是在极少数情况下进行的。作为项目的一部分，我计划1）研究分次Brauer图代数B_r的球面扭沿着投射模生成的自等价群的子群，并证明这个子群与曲面B_r的辫子扭群同构; 2）研究分次Brauer图代数的内禀形式，推广（1）中的结果到任何增强的三角形类别与适当配置的球形物体（从理论上讲，这类范畴作为某些福谷范畴出现，并在聚类范畴的研究中出现）;第三章利用映射类的某些子群得到Brauer图代数的有界导出范畴的自等价群的完整描述B_r的群，B_r的移位和外自同构.

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.