Preprojective algebras of tree-type quivers

Abstract

Let $\mathbf{k}Q$ be the path algebra of a tree-type quiver Q, and λ be a nonzero element in a 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 λ-relations. When λ = 1, the relations are known as mesh relations. When $\lambda =-1$, they are known as commutativity relations. We give a new description of the preprojective algebra of $\mathbf{k}Q$ 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.

Keywords:

• Derived category
• preprojective algebras
• quivers
• standard
• tree

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 16G20
• 16G70

Acknowledgements

The authors thank Bill Crawley–Boevey for insightful discussions and for pointing out several relevant results on preprojective algebras. The authors also thank the referee for useful comments, especially for encouraging us to generalize our main result about the algebra isomorphism for all $\lambda \in {\mathbf{k}}^{\times }$. This work was done when the first author was a Zelevinsky Research Instructor at Northeastern University; she thanks the Mathematics Department for their support.