On the structure of triangulated categories with finitely many indecomposables

Abstract

We study the problem of classifying triangulated categories with finite-dimensional morphism spaces and finitely many indecomposables over an algebraically closed field k{script}. We obtain a new proof of the following result due to Xiao and Zhu: the Auslander-Reiten quiver of such a category T{script} is of the form Z{script}Δ/G{script} where Δis a disjoint union of simply-laced Dynkin diagrams and G{script} a weakly admissible group of automorphisms of Z{script}Δ. Then we prove that for 'most' groups G{script}, the category T{script} is standard, i.e. K{script}-linearly equivalent to an orbit category D{script}b(mod k{script}Δ)/Ψ. This happens in particular when T{script} is maximal d-Calabi-Yau with d≥2. Moreover, if T{script} is standard and algebraic, we can even construct a triangle equivalence between T and the corresponding orbit category. Finally we give a sufficient condition for the category of projectives of a Frobenius category to be triangulated. This allows us to construct non standard 1-Calabi-Yau categories using deformed preprojective algebras of generalized Dynkin type. ©SMF.