Sur la catégorie amassée d'une surface marquée
We study in this thesis the cluster category C[subscript S,M] and cluster algebra A[Subscript S,M] of a marked surface (S, M) without punctures. We give a geometric characterization of the indecomposable objects in C[subscript S,M] as homotopy classes of curves in (S, M) and one-parameter families related to non-contractible closed curves in (S , M). Moreover, the Auslander-Reiten structure of the category C[Subscript S,M] is described in geometric terms and we show that the objects without self-extensions in C[Subscript S,M] correspond to curves in (S, M) without self-intersections. As a consequence, we establish that every rigid indecomposable object is reachable from an initial triangulation. As for the cluster algebra A[Subscript S,M], we give a module-theoretic interpretation of Schiffler's expansion formula defined in . Based on the properties of the cluster category C[Subscript S,M], we show the coincidence of Schiffler-Thomas' expansion formula and the cluster character defined in .
- Sciences – Thèses