$\mathbb Z Q$ type constructions in higher representation theory

Abstract

Let $Q$ be an acyclic quiver, it is classical that certain truncations of the translation quiver $\mathbb Z Q$ appear in the Auslander-Reiten quiver of the path algebra $kQ$. The stable $n$-translation quiver $\mathbb Z|_{n-1} Q$ is introduced as a generalization of the $\mathbb Z Q$ construction in studying higher representation theory of algebras for an acyclic bound quiver $Q$. In this paper, we find conditions for a Hom-finite Krull-Schmidt $k$-category to be realized as the bound path category of a convex full subquiver of an stable $n$-translation quiver.We show that for $n$-slice algebra $\Gamma$, which is an $n$-hereditary algebra whose $(n+1)$-preprojective algebra is $(q+1,n+1)$-Koszul, with bound quiver $Q^{op}$, its $n$-preprojective and $n$-preinjective components in the module category and truncations of the stable $n$-translation quiver $\mathbb Z|_{n-1} Q^{op}$. We also use $\mathbb Z|_{n-1} Q^{op}$ to describe the $\nu_n$-closure of $\Gamma$ in the derived category.