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

TL;DR

This paper establishes a geometric correspondence between indecomposable modules over a specific affine quiver and planar curves, linking self-intersections and intersections to extension space dimensions in module and cluster categories.

Contribution

It introduces a novel bijection between modules and planar curves for affine type quivers, connecting geometric intersections with algebraic extension dimensions.

Findings

Number of self-intersections equals the dimension of Ext^1(M,M).

Curve intersections correspond to Ext^1(M,N) dimensions in the cluster category.

Provides a geometric model for understanding module extensions in affine quivers.

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 $\text{Ext}^1_{kQ}(M,M)$. 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 $\text{Ext}^1_C (M,N)$, where $C$ is the cluster category of $kQ$-mod.