On the representation type of higher preprojective algebras of type $A$

Abstract

In this paper, we establish a finite--tame--wild trichotomy for the representation type of $(n+1)$-preprojective algebras of type $A_m$. We also provide a complete classification of such algebras with respect to $τ$-tilting finiteness. Moreover, for $n\ge 2$, we give a precise characterization of when each indecomposable module is $τ$-rigid.