Computation of extension spaces for the path algebra of type $\tilde{A}(n-1,1)$ using planar curves

Abstract

$Q$ is a quiver of type $\tilde{A}(n-1,1)$ if its graph is of affine type ${\tilde{A}}_{n-1}$ and if its arrows have a certain orientation. We develop a bijection between the set of indecomposable $kQ$-modules whose dimension vectors are positive real roots of the root system associated to $Q$ and a certain set of planar curves. We prove that the number of self-intersections of the curve which corresponds to the module $M$ is equal to the dimension of ${\text{Ext}}_{kQ}^1(M,M)$. We also prove that, for many pairs of modules $(M,N)$, the number of intersections between the corresponding two curves is equal to the dimension of ${\text{Ext}}_C^1(M,N)$, where $C$ is the cluster category of $kQ$-mod.

v2: references and acknowledgments updated, more details added to Background section, overall exposition improved

References in corpus (2)

• Cluster categories for marked surfaces: punctured case
• Acyclic cluster algebras, reflection groups, and curves on a punctured disc