Preprojective algebras of tree-type quivers

Abstract

Let be the path algebra of a tree-type quiver Q, and λ be a nonzero element in a field . We construct irreducible morphisms in the Auslander–Reiten quiver of the transjective component of the bounded derived category of that satisfy what we call the λ-relations. When λ = 1, the relations are known as mesh relations. When , they are known as commutativity relations. We give a new description of the preprojective algebra of and using our technique of constructing irreducible maps together with the results given by Baer–Geigle–Lenzing, Crawley–Boevey, Ringel, and others, we show that for any tree-type quiver, our description is equivalent to several other definitions of preprojective algebras, previously introduced in various contexts.