Preprojective algebras of tree-type quivers

Let be the path algebra of a tree-type quiver Q, and lambda 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 lambda-relations. When lambda = 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.