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

• Applicant: Alexandra Zvonareva, Ph.D.
• Subject Area: Mathematics
• Term: from 2022 to 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 512295948

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.

• DFG Programme: Research Grants