A geometric model for the bounded derived category of a gentle algebra 1

Abstract: Gentle algebras are quadratic monomial algebras whose representation theory is well understood. In recent years they have played a central role in several different subjects such as in cluster algebras where they occur as Jacobian algebras of quivers with potentials obtained from triangulations of marked surfaces and in the context of homological mirror symmetry where graded gentle algebras with zero differential appear in the construction of partially wrapped Fukaya categories of surfaces with stops. In these lectures we will recall the construction of a geometric model for the bounded derived category of an (ungraded) gentle algebra. We will see how the gentle algebra encodes not only a marked surface but also a line field on the surface. This line field allows to define a complete derived invariant for gentle algebras which generalises and completes the well-known derived invariant by Avella-Alaminos and Geiss. We will give explicit examples relating the introduced geometric model with the description of the partially wrapped Fukaya category in the work of Haiden, Katzarkov and Kontsevich.

rings and algebrasrepresentation theorysymplectic geometry

Audience: researchers in the discipline

Winter School "Connections between representation theory and geometry"

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