Total preprojective algebras

TL;DR

This paper introduces total preprojective algebras for Dynkin quivers, proves their isomorphism to 2-Auslander algebras, and describes their structure and presentations, extending to higher dimensions.

Contribution

It defines total preprojective algebras, establishes their isomorphism with 2-Auslander algebras, and provides explicit descriptions and presentations, including higher-dimensional generalizations.

Findings

Total preprojective algebras are isomorphic to 2-Auslander algebras.

They have global dimension 3 and dominant dimension 3.

Explicit quiver presentations of these algebras are provided.

Abstract

We introduce total preprojective algebras $\Psi$ of path algebras of Dynkin quivers $kQ$, and prove that they are isomorphic to $2$-Auslander algebras of preprojective algebras $\Pi$ of $kQ$. In particular, $\Psi$ has global dimension $3$ and dominant dimension $3$. We also describe $\Psi$ as a tensor algebra of a certain explicit bimodule over the Auslander algebra of $kQ$. As an application, we give a presentation of $\Psi$ by explicit quivers with relations. More generally, we introduce total $(d+1)$-preprojective algebras of $d$-representation finite algebras, and give all the corresponding results.