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Auslander algebras and initial seeds for cluster algebras

Christof Geiß, Bernard Leclerc, Jan Schröer

Number 34
Author Jan Schröer
Year 2007
Journal Journal of the London Mathematical Society (2) 75 (2007), 718-740.

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 IQ 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  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, we exploit the fact that the categories of injective modules over Λ and over its covering Λ are triangulated in order to show several interesting identities in the respective stable module categories.

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