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, a rigid Λ-module IQ is produced with r = |Π| pairwise non-isomorphic indecomposable direct summands by pushing the injective modules of the Auslander algebra of k Q 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. It is shown that the elements of the dual semicanonical basis which correspond to the indecomposable direct summands of IQ 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 EndΛ(IQ). Finally, the fact that the categories of injective modules over Λ and over its covering Λ̃ are triangulated is exploited in order to show several interesting identities in the respective stable module categories.