A reflection equivalence for Gorenstein-projective quiver representations

Abstract

For Λ a selfinjective algebra, and Q a finite quiver without oriented cycles, the algebra ΛQ is a Gorenstein algebra and the category G-proj ΛQ of Gorenstein-projective ΛQ-modules is a Frobenius category. For a sink v of Q, we define a functor F (v) : G-proj ΛQ → G-proj ΛQ(v) between the stable categories modulo projectives, where Q(v) is obtained from Q by changing the direction of each arrow ending in v. The functor is given by an explicit construction on the level of objects and homomorphisms. Our main result states that F (v) is an equivalence of categories. In the case where the underlying graph of Q is a tree, we deduce that the stable category G-proj ΛQ does not depend on the orientation of Q. Moreover, if Q is a quiver of type A 3 and Λ = k[T ]/(T n ) the bounded polynomial algebra, we use the symmetry of the octahedron in the octahedral axiom to verify that the composition of twelve reflections yields the identity on objects.MSC 2020: 18G25 (relative homological algebra), 16G20 (representations of quivers)