Dimension vectors of elementary modules of generalized Kronecker quivers

Abstract

Let $k$ be an algebraically closed field. The generalized or $n$-Kronecker quiver $K(n)$ is the quiver with two vertices, called a source and a sink, and $n$ arrows from source to sink. Given a finite-dimensional module $M$ of the path algebra $kK(n)={\mathcal{K}}_n$, we consider its dimension vector $\underline{dim}M=({\dim }_k{M}_1,{\dim }_k{M}_2)$. Let $\mathbf{F}=\{(x,y)\mid \frac{2}{n}x\le y\le x\}$, and let $(x,y)\in \mathbf{F}$. We construct a module $X(x,y)$ of ${\mathcal{K}}_n$, and we prove it to be elementary. Suppose that $\underline{dim}M=(x,y)$. We show that: if $M$ is an elementary module, then $x<2n$, and when $x+y=n+1$, the module $M$ is elementary if and only if $M$ is of the form $X(x,y)$.