arXiv subjects
Publications and source records attributed to Jin Yun Guo.
In this paper we show that acyclic $n$-slice algebras are exactly acyclic $n$-hereditary algebras whose $(n+1)$-preprojective algebras are $(q+1,n+1)$-Koszul. We also list the equivalent triangulated categories arising from the algebra constructions related to an $n$-slice algebra. We show that higher slice algebras of finite type appear in pairs and they share the Auslander-Reiten quiver for their higher preprojective components.
In this paper, we draw diagrams of yinyang, wuxing and bagua as the McKay quivers of the groups of order 2, of order $5$ and of order $2^3$.
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.
The $n$-slice algebra is introduced as a generalization of path algebra in higher dimensional representation theory. In this paper, we give a classification of $n$-slice algebras via their $(n+1)$-preprojective algebras and the trivial extensions of their quadratic duals. One can always relate tame $n$-slice algebras to the McKay quiver of a finite subgroup of $\mathrm{GL}(n+1, \mathbb C)$. In the case of $n=2$, we describe the relations for the $2$-slice algebras related to the McKay quiver of finite Abelian subgroups of $\mathrm{SL}(3, \mathbb C)$ and of the finite subgroups obtained from embedding $\mathrm{SL}(2, \mathbb C)$ into $\mathrm{SL}(3,\mathbb C)$.
APR tilts for path algebra $kQ$ can be realized as the mutation of the quiver $Q$ in $\mathbb Z Q$ with respect to the translation. In this paper, we show that we have similar results for the quadratic dual of truncations of $n$-translation algebras, that is, under certain condition, the $n$-APR tilts of such algebras are realized as $τ$-mutations.For the dual $τ$-slice algebras with bound quiver $Q^{\perp}$, we show that their iterated $n$-APR tilts are realized by the iterated $τ$-mutations in $\mathbb Z|{n-1}Q^{\perp}$.
In this paper, we show that for a Koszul $n$-homogeneous algebra $Λ$, the quadratic dual of certain twisted trivial extension is the $(n+1)$-preprojective algebra of its quadratic dual, that is, $ (Δ_νΛ)^{!,op} \simeqΠ( Λ^{ !, op })$. This is applied to the $τ$-slice algebras of stable $n$-translation algebras and gives a noncommutative version of Bernstein-Gelfand-Gelfand correspondence for such algebras.
In this paper, we show that the $n$-APR tilts of dual $τ$-slice algebras of acyclic stable $n$-translation algebras are realized as $τ$-mutations. Such dual $τ$-slice algebras are quasi $(n-1)$-Fano when the $n$-translation algebra is Koszul, and a recursive construction of higher quasi Fano algebras for quasi $n$-Fano algebra obtained in this way is given. The $τ_n$-closure and $ν_n$-closure of such algebras are studied and we show that for an acyclic dual $n$-translation algebras with bound quiver $Q^{\perp}$, the Auslander-Reiten quivers of its $τ_n$-closures are truncation of the quiver $\mathbb{Z}|_n Q^{\perp}$, and the Auslander-Reiten quiver of its $ν_n$-closure is $\mathbb{Z}|_n Q^{\perp}$ when it is $n$-representation infinite.
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.
Motivated by Iyama's higher representation theory, we introduce $n$-translation quivers and $n$-translation algebras. The classical $\mathbb Z Q$ construction of the translation quiver is generalized to construct an $(n+1)$-translation quiver from an $n$-translation quiver, using trivial extension and smash product. We prove that the quadratic dual of $n$-translation algebras have $(n-1)$-almost splitting sequences in the category of its projective modules. We also present a non-Koszul $1$-translation algebra whose trivial extension is $2$-translation algebra, thus also provides a class of examples of $(3,m-1)$-Koszul algebras (and also a class of $(m-1,3)$-Koszul algebras) for all $m \ge 2$.
In this paper, we discuss returning arrows with respect to the Nakayama translation appearing in the quivers of some important algebras when we construct extensions. When constructing twisted trivial extensions for a graded self-injective algebra, we show that the returning arrows appear in the quiver, that the complexity increases by 1 in Koszul cases, and the representation dimension also increases by 1 under certain additional conditions. By applying Koszul duality, for each Koszul Artin-Schelter regular algebra of global dimension l and Gelfand-Kirilov dimension $c$, we construct a family of Koszul Artin-Schelter regular algebras of global dimension $l+1$ and Gelfand-Kirilov dimension $c+1$, among them one is central extension and one is $l+1$-Calabi-Yau.
Let $Λ$ be a graded self-injective algebra. We describe its smash product $Λ# k\mathbb Z^*$ with the group $\mathbb Z$, its Beilinson algebra and their relationship. Starting with $Λ$, we construct algebras with finite global dimension, called $τ$-slice algebras, we show that their trivial extensions are all isomorphic, and their repetitive algebras are the same $Λ# k\mathbb Z^*$. There exist $τ$-mutations similar to the BGP reflections for the $τ$-slice algebras. We also recover Iyama's absolute $n$-complete algebra as truncation of the Koszul dual of certain self-injective algebra.
In this paper, we study the relationship between the McKay quivers of a finite subgroups $G$ of special linear groups general linear groups, via some natural extension and embedding. We show that the McKay quiver of certain extension of a finite subgroup $G$ of $\mathrm{SL}(m,\mathbb C)$ in $\mathrm{GL}(m,\mathbb C)$ is a regular covering of the McKay quiver of $G$, and when embedding $G$ in a canonical way into $\mathrm{GL}(m-1,\mathbb C)$, the new McKay quiver is obtained by adding an arrow from the Nakayama translation of $i$ back to $i$ for each $i$. We also show that certain interesting examples of McKay quivers are obtained in these two ways.