Auslander algebras and initial seeds for cluster algebras

By C. Geiss, Bernard Leclerc, J. Schröer. 2005

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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.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/f04621cf8c5304171b7d0ba69a889e5eca124f07

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Citations in the corpus

Cluster structures for 2-Calabi–Yau categories and unipotent groups (2007, methodology): “ws X→τXwhen Xis nonprojective in P, we get the quiver of the cluster tilting object given by the above reduced expression. That this quiver is the quiver of a cluster tilting object was also shown in [GLS3] for Pbeing the AR quiver of a Dynkin quiver, and in [GLS6] in the general case. LemmaII.4.4. ”
Representations of Algebras and Related Topics (2011, background): “k between cluster algebras and preprojective algebras can be found in [47], [48], [50], [51] (see also [49] for an overview). On generalized cluster categories 15 Theorem 2.11 (Geiss-Leclerc-Schr¨oer [46]). Let ∆ be a simply laced Dynkin diagram. Then for any orientation Q of ∆, there exists a clust”
n-Abelian and n-exact categories (2014, background): “It is known that the standard 2-cluster-tilting Λ-module T corresponding to the linear orientation of An satisfies ℧ (2)(T ) ∼= T , see [21].”
Acyclic Calabi–Yau categories (2006, background): “The question arises as to whether, for a given quiver Q without oriented cycles, the cluster category is the ‘unique model’ of the associated cluster algebra.”
Cluster algebras and representation theory (2010, background): “the quiver with vertex set {1,...,r}and dij arrows from i to j, where dij = dimExt1 B(si,sj). Theorem 6.1([23]). The quiver ΓT has no loops nor 2-cycles. Deﬁne Σ(T) = ((ϕT1,...,ϕTr ), ΓT). Theorem6.2([24]). There exists an explicit maximal rigid Λ-module U such that Σ(U) is one of the seeds of the c”