arXiv subjects
Publications and source records attributed to Deren Luo.
In this paper, we introduce multi-layer quiver and show how to construct an $(n+1)$-slice algebras of infinite type from an $n$-slice algebra of infinite type using the bound quivers. This leads to constructing $(n+1)$-slice algebras of infinite type as matrix algebra and as tensor algebra of an $n$-slice algebra and equivalences of their module categories as the module categories of diagram of some quiver of type $\widetilde{A}_{n+1}$.
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 $Γ$, 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 $ν_n$-closure of $Γ$ in the derived category.
In this note, we show that the long $n$-exact sequence theorem holds in any $n$-abelian category. As an application, the $n$-abelian version of Wakamatsu's Lemma is given.
In this paper we study a class of algebras having $n$-dimensional pyramid shaped quiver with $n$-cubic cells, which we called $n$-cubic pyramid algebras. This class of algebras includes the quadratic dual of the basic $n$-Auslander absolutely $n$-complete algebras introduced by Iyama. We show that the projective resolution of the simples of $n$-cubic pyramid algebras can be characterized by $n$-cuboids, and prove that they are periodic. So these algebras are almost Koszul and $(n-1)$-translation algebras. We also recover Iyama's cone construction for $n$-Auslander absolutely $n$-complete algebras using $n$-cubic pyramid algebras and the theory of $n$-translation algebras.
Let $Γ^{n}$ be the cone of an $(n-1)$-complete algebra over an algebraically closed field $k$. In this paper, we prove that if the bound quiver $(Q_{n},ρ_{n})$ of $Γ^{n}$ is a truncation from the bound McKay quiver $(Q_{G},ρ_{G})$ of a finite subgroup $G$ of $GL(n,k)$, the bound quiver $(Q_{n+1}, ρ_{n+1})$ of $Γ^{n+1}$, the cone of $Γ^{n}$, is a truncation from the bound McKay quiver $(Q_{\widetilde{G}},ρ_{\widetilde{G}})$ of $\widetilde{G}$, where $\widetilde{G}\cong G\times \mathbb{Z}_{m}$ for some $m\in \mathbb{N}$.