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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 $kQ$. We introduce the $n$-translation quiver $mathbb Z|_{n-1} Q$ as a generalization of the $mathbb Z Q$ construction in our recent study on $n$-translation algebras. In this paper, we introduce $n$-slice algebra and show that for certain $n$-slice algebra $Gamma$, % of global dimension $n$, the quiver $mathbb Z|_{n-1} Q$ can be used to describe the $tau_n$-closure of $DGamma$ and $tau_n^{-1}$-closure of $Gamma$ in its module category and the $ u_n$-closure of $Gamma$ in the derived category.
The canonical bases of cluster algebras of finite types and rank 2 are given explicitly in cite{CK2005} and cite{SZ} respectively. In this paper, we will deduce $mathbb{Z}$-bases for cluster algebras for affine types $widetilde{A}_{n,n},widetilde{D}$
This paper reports some advances in the study of the symplectic blob algebra. We find a presentation for this algebra. We find a minimal poset for this as a quasi-hereditary algebra. We discuss how to reduce the number of parameters defining the alge
For a commutative algebra $A$ over $mathbb{C}$,denote $mathfrak{g}=text{Der}(A)$. A module over the smash product $A# U(mathfrak{g})$ is called a jet $mathfrak{g}$-module, where $U(mathfrak{g})$ is the universal enveloping algebra of $mathfrak{g}$.In
For a complex finite-dimensional simple Lie algebra $mathfrak{g}$, we introduce the notion of Q-datum, which generalizes the notion of a Dynkin quiver with a height function from the viewpoint of Weyl group combinatorics. Using this notion, we develo
In this note, we consider the $d$-cluster-tilted algebras, the endomorphism algebras of $d$-cluster-tilting objects in $d$-cluster categories. We show that a tilting module over such an algebra lifts to a $d$-cluster-tilting object in this $d$-cluster category.