G-semisimple algebras

Abstract

Let $Λ$ be an Artin algebra and ${\mathsf{mod}}\mbox{-} ({\underline{\mathsf{Gprj}}}\mbox{-}Λ)$ the category of finitely presented functors over the stable category ${\underline{\mathsf{Gprj}}}\mbox{-}Λ$ of finitely generated Gorenstein projective $Λ$-modules. This paper deals with those algebras $Λ$ in which ${\mathsf{mod}}\mbox{-} ({\underline{\mathsf{Gprj}}}\mbox{-}Λ)$ is a semisimple abelian category, and we call G-semisimple algebras. We study some basic properties of such algebras. In particular, it will be observed that the class of G-semisimple algebras contains important classes of algebras, including gentle algebras and more generally quadratic monomial algebras. Next, we construct an epivalence from the stable category of Gorenstein projective representations $\underline{\mathsf{Gprj}}(\mathcal{Q}, Λ)$ of a finite acyclic quiver $\mathcal{Q}$ to the category of representations ${\rm rep}(\mathcal{Q}, \underline{\mathsf{Gprj}}\mbox{-} Λ)$ over $\underline{\mathsf{Gprj}}\mbox{-} Λ)$, provided $Λ$ is a G-semisimple algebra over an algebraic closed field. Using this, we will show that the path algebra $Λ\mathcal{Q}$ of the G-semisimple algebra $Λ$ is Cohen-Macaulay finite if and only if $\mathcal{Q}$ is Dynkin. In the last part, we provide a complete classification of indecomposable Gorenstein projective representations within ${\mathsf{Gprj}}(A_n, Λ)$ of the linear quiver $A_n$ over a G-semisimple algebra $Λ$. We also determine almost split sequences in ${\mathsf{Gprj}}(A_n, Λ)$ with certain ending terms. We apply these results to obtain insights into the cardinality of the components of the stable Auslander-Reiten quiver ${\mathsf{Gprj}}(A_n, Λ)$.