Graded Kronecker Modules of Constant Rank

Abstract

Let $K(n)$ be the generalized Kronecker quiver, and let $(T(n),\Omega)$ be the universal covering of $K(n)$, where $n\geq 3$ and $\Omega$ is a bipartite orientation. We introduce three different types of representations for quiver $(T(n),\Omega)$ by $K(n)$: maps are injective in a representation maps are all surjective in a representation, and representations are of constant rank. We use $\Inj, \Sur$ and $\GR$ to denote the corresponding subcategories containing these representations, respectively, and we show that \begin{center} $\GR= \Inj\cup\Sur$. \end{center}