2019/08/19 by Guo, Jin Yun, Lu, Xiaojian, Luo, Deren
#16G20 #16G70 #16S37 #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.1908.06546
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 Qop, its n-preprojective and n-preinjective components in the module category and truncations of the stable n-translation quiver \mathbb Z|n-1 Qop. We also use \mathbb Z|n-1 Qop to describe the νn-closure of Γ in the derived category.