Elementary modules of the generalized Kronecker quivers K(n)

RESUMO

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. We use $\modd\cK_n$ to denote the category of the finite-dimensional modules of the path algebra $\cK_n=kK(n)$. There exist a function called dimension vector $\dimu: \modd \cK_n\rightarrow \mathbb{Z}^2; M \mapsto (\dim_k M_1, \dim_k M_2 )$, and two functors $\sigma: \mathbb{Z}^2 \rightarrow \mathbb{Z}^2; (x,y)\mapsto (nx-y,x)$, and $\delta: \mathbb{Z}^2\rightarrow \mathbb{Z}^2;(x,y)\mapsto (y,x)$. Then the set $\mathbf{F}=\{(x,y)\mid \frac{2}{n}x\leq y\leq x\}$ is the fundamental domain of the dimension vectors of regular modules under the action of $\delta$ and $\sigma$ in $\modd\cK_n$. We call a regular module $M\in \modd\cK_n$ \textit{elementary} if there does not exist a short exact sequence $(0)\rightarrow L\rightarrow M\rightarrow N\rightarrow (0)$ with $L,N$ being non-zero regular. In this note, we focus on the set $\mathbf{F}$, and when $x<n$, we establish a one-to-one correspondence between elementary modules and spaces of fixed rank.