Preprojective algebras of tree-type quivers

TL;DR

This paper constructs specific irreducible morphisms satisfying lambda-relations in the derived category of tree-type quivers and demonstrates the equivalence of multiple descriptions of their preprojective algebras.

Contribution

It introduces the concept of lambda-relations for irreducible morphisms in the derived category of tree-type quivers and proves the equivalence of various descriptions of their preprojective algebras.

Findings

Construction of irreducible morphisms satisfying lambda-relations

Establishment of equivalence among different descriptions of preprojective algebras

Extension of known mesh and commutativity relations to lambda-relations

Abstract

Let $Q$ be a tree-type quiver, $\mathbf{k} Q$ its path algebra, and $\lambda$ a nonzero element in the field $\mathbf{k}$. We construct irreducible morphisms in the Auslander-Reiten quiver of the transjective component of the bounded derived category of $\mathbf{k} Q$ that satisfy what we call the $\lambda$-relations. When $\lambda=1$, the relations are known as mesh relations. When $\lambda=-1$, they are known as commutativity relations. Using this technique together with the results given by Baer-Geigle-Lenzing, Crawley-Boevey, Ringel, and others, we show that for any tree-type quiver, several descriptions of its preprojective algebra are equivalent.