Spherical objects on cycles of projective lines and transitivity

Sebastian Opper (Charles University Prague, Czech Republic)

04-Mar-2021, 12:00-13:00 (3 years ago)

Abstract: Polishchuk showed that spherical objects in the derived category of any cycle of projective lines yield solutions of the associative Yang-Baxter equation which raises the question whether one can classify spherical objects. He further posed the question whether the group of derived auto-equivalences of a cycle acts transitively on isomorphism classes of spherical objects. Partial solutions to both problems were given in works of Burban-Kreussler and Lekili-Polishchuk. A theorem of Burban-Drozd establishes a connection between the derived category of any cycle of projective lines with the derived category of a certain gentle algebra which can be modeled by a (toplogical) surface and which allows us to translate algebraic information in the derived category such as objects into geometric information on the surface such as curves. I will explain how the result of Burban-Drozd can be used to find a similar model for the derived category of a cycle. Afterwards we discuss how this can be exploited to classify spherical objects and establish transitivity. Further applications include a description of the group of derived auto-equivalences of a cycle and faithfulness of a certain group action as defined by Sibilla.

mathematical physicscommutative algebraalgebraic geometryrings and algebrasrepresentation theorysymplectic geometry

Audience: researchers in the topic

( slides | video )


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