Speaker
Description
The derived category of a gentle algebra A can be described via the geometry of surface dissections, where indecomposable complexes of A-modules correspond to possibly infinite graded arcs on the surface, and morphisms between them are encoded by crossings between the associated arcs. Moreover, their mapping cones are given by the resolution of these crossings.
We introduce a Caldero-Chapoton map in this setting. Specifically, we associate a Laurent polynomial to every finite indecomposable complex, and show that skein relations hold whenever the corresponding arcs cross in the interior. For each complex, its Caldero-Chapoton map specializes to the corresponding element of the Grothendieck group. Furthermore, if A is hereditary, the algebra generated by these functions is an ordinary cluster algebra.
This is joint work in progress with Esther Banaian, Ilaria Di Dedda, Khrystyna Serhiyenko, Yadira Valdivieso-Díaz and Kayla Wright.