Total preprojective algebras

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