Total preprojective algebras

Abstract

We introduce total preprojective algebras $\hat{I}¨$ of path algebras of Dynkin quivers $kQ$, and prove that they are isomorphic to $2$-Auslander algebras of preprojective algebras $\hat{I}$ of $kQ$. In particular, $\hat{I}¨$ has global dimension $3$ and dominant dimension $3$. We also describe $\hat{I}¨$ as a tensor algebra of a certain explicit bimodule over the Auslander algebra of $kQ$. As an application, we give a presentation of $\hat{I}¨$ 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.