G-semisimple algebras

Abstract

Let $\Lambda$ 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 $\Lambda$-modules. This paper deals with those algebras $\Lambda$ 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 $\underline{\mathsf{Gprj}}(\mathcal{Q},\Lambda )$ 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 $\Lambda$ is a G-semisimple algebra over an algebraic closed field. Using this, we will show that the path algebra $\Lambda \mathcal{Q}$ of the G-semisimple algebra $\Lambda$ 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 $\mathsf{Gprj}(A_n,\Lambda )$ of the linear quiver $A_n$ over a G-semisimple algebra $\Lambda$. We also determine almost split sequences in $\mathsf{Gprj}(A_n,\Lambda )$ with certain ending terms. We apply these results to obtain insights into the cardinality of the components of the stable Auslander-Reiten quiver $\mathsf{Gprj}(A_n,\Lambda )$.

This paper is an enhanced version of the paper with arXiv identifier:2109.00467. A correction has been made in Section 4