G-semisimple algebras

Let Λ be an Artin algebra and $\mathrm{mod}\text{-}\left(\underset{\_}{\mathrm{Gprj}}\text{-}\Lambda \right)$ the category of finitely presented functors over the stable category $\underset{\_}{\mathrm{Gprj}}\text{-}\Lambda$ of finitely generated Gorenstein projective Λ-modules. This paper deals with those algebras Λ in which $\mathrm{mod}\text{-}\left(\underset{\_}{\mathrm{Gprj}}\text{-}\Lambda \right)$ 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 (called representation equivalence in the terminology of Auslander), i.e. a full and dense functor that reflects isomorphisms, from the stable category of Gorenstein projective representations $\underset{\_}{\mathrm{Gprj}}(Q,\Lambda )$ of a finite acyclic quiver $Q$ to the category of representations $\mathrm{rep}(Q,\underset{\_}{\mathrm{Gprj}}\text{-}\Lambda )$ over $\underset{\_}{\mathrm{Gprj}}\text{-}\Lambda$, provided Λ is a G-semisimple algebra over an algebraic closed field. Using this, we will show that the path algebra $\Lambda Q$ of the G-semisimple algebra Λ is $\mathrm{CM}$-finite if and only if $Q$ is Dynkin. In the last part, we provide a complete classification of indecomposable Gorenstein projective representations within $\mathrm{Gprj}(A_n,\Lambda )$ of the linear quiver $A_n$ over a G-semisimple algebra Λ. We also determine almost split sequences in $\mathrm{Gprj}(A_n,\Lambda )$ with certain ending terms. We apply these results to obtain insights into the cardinality of the components of the stable Auslander-Reiten quiver $\mathrm{Gprj}(A_n,\Lambda )$.