Konfigurationen von sphärischen Twists und derivierte Picardgruppen von Brauergraphalgebren
Zusammenfassung der Projektergebnisse
Derived categories and other triangulated categories are an indispensable tool in representation theory, algebraic geometry, topology, and other areas of mathematics. In representation theory of finite dimensional algebras they are one of the main objects of study. This project focuses on studying symmetries of derived categories of a certain family of algebras appearing in representation theory, namely Brauer graph algebras and their graded analogues. Brauer graph algebras appeared in representation theory as a generalization of blocks with cyclic and dihedral defect group, their category of modules is very well understood. A Brauer graph algebra BΓ can be constructed from a Brauer graph Γ which is a graph embedded into an oriented surface ΣΓ in a minimal way (in this case ΣΓ is unique up to homeomorphism) together with a multiplicity function which assigns a non-zero natural number to each vertex. The surface ΣΓ and the connection between Brauer graph algebras and gentle algebras established by Schroll allows to relate Brauer graph algebras to Fukaya categories of surfaces with stops arising in symplectic geometry. In the previous work Opper-Zvonareva used this connection to obtain a complete classification of Brauer graph algebras under derived equivalences. Brauer graph algebras exhibit interesting behaviour from the point of view of derived Picard groups or groups of autoequivalences of the derived categories. Projective modules over this class of algebras are examples of spherical objects, special class of objects which can be used to construct autoequivalences of the derived category, called spherical twists. Spherical twists were used to construct braid group actions on various triangulated categories and study their faithfulness. The first aim of this project was to construct and study the action of the braid twist group of the surface ΣΓ with punctures corresponding to the vertices of Γ (that is the group generated by all half-twists along curves, connecting two punctures) on the derived category of the algebra BΓ using the action of the spherical twists corresponding to the projective modules over BΓ. During the first year of the project I concentrated mainly on this goal.
Projektbezogene Publikationen (Auswahl)
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Derived equivalence classification of Brauer graph algebras. Advances in Mathematics, 402, 108341.
Opper, Sebastian & Zvonareva, Alexandra
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Lifting and restricting t‐structures. Bulletin of the London Mathematical Society, 55(2), 640-657.
Marks, Frederik & Zvonareva, Alexandra
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Co-t-structures, cotilting and cotorsion pairs. Mathematical Proceedings of the Cambridge Philosophical Society, 175(1), 89-106.
PAUKSZTELLO, DAVID & ZVONAREVA, ALEXANDRA
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Lattices of t‐structures and thick subcategories for discrete cluster categories. Journal of the London Mathematical Society, 107(3), 973-1001.
Gratz, Sira & Zvonareva, Alexandra
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A functorial approach to rank functions on triangulated categories. Journal für die reine und angewandte Mathematik (Crelles Journal), 0(0).
Conde, Teresa; Gorsky, Mikhail; Marks, Frederik & Zvonareva, Alexandra
