G-semisimple algebras

Let A be an Artin algebra and mod -( Gprj -A ) the category of finitely presented functors over the stable category Gprj -A of finitely generated Gorenstein projective A-modules. This paper deals with those algebras A in which mod -( Gprj -A ) 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 Gprj (Q, A) of a finite acyclic quiver Q to the category of representations rep(Q, Gprj -A ) over Gprj -A , provided A is a G-semisimple algebra over an algebraic closed field. Using this, we will show that the path algebra AQ of the G-semisimple algebra A is CM-finite if and only if Q is Dynkin. In the last part, we provide a complete classification of indecomposable Gorenstein projective representations within Gprj(A n , A) of the linear quiver A n over a G-semisimple algebra A. We also determine almost split sequences in Gprj(A n , A) with certain ending terms. We apply these results to obtain insights into the cardinality of the components of the stable Auslander-Reiten quiver Gprj (A n , A). (c) 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.