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 $\mathbf{F}=\{(x,y)\mid \frac{2}{n}x\leq y\leq x\}$, 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 $X(x,y)$.