Auslander algebras and initial seeds for cluster algebras

Abstract

Let $Q$ be a Dynkin quiver and $Π$ the corresponding set of positive roots. For the preprojective algebra $Λ$ associated to $Q$ we produce a rigid $Λ$-module $I_Q$ with $r=|Π|$ pairwise non-isomorphic indecomposable direct summands by pushing the injective modules of the Auslander algebra of $kQ$ to $Λ$. If $N$ is a maximal unipotent subgroup of a complex simply connected simple Lie group of type $|Q|$, then the coordinate ring $C[N]$ is an upper cluster algebra. We show that the elements of the dual semicanonical basis which correspond to the indecomposable direct summands of $I_Q$ coincide with certain generalized minors which form an initial cluster for $C[N]$, and that the corresponding exchange matrix of this cluster can be read from the Gabriel quiver of $End_Λ(I_Q)$. Finally, we exploit the fact that the categories of injective modules over $Λ$ and over its covering $\tildeΛ$ are triangulated in order to show several interesting identities in the respective stable module categories.