Rigid modules over preprojective algebras

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

256 citations recorded by the source corpus.

Let Λ be a preprojective algebra of simply laced Dynkin type Δ. We study maximal rigid Λ-modules, their endomorphism algebras and a mutation operation on these modules. This leads to a representation-theoretic construction of the cluster algebra structure on the ring ℂ[N] of polynomial functions on a maximal unipotent subgroup N of a complex Lie group of type Δ. As an application we obtain that all cluster monomials of ℂ[N] belong to the dual semicanonical basis.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c5ef27244a920361cfef3b486085ab29eaec51ff

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Citations in the corpus

Cluster algebras and derived categories (2012, unknown): “A proof of the corollary based on the study of stability conditions on Dfd(Γ) is given in section 4.2 of [106].”
Cluster structures on strata of flag varieties (2014, methodology): “it short exact sequences 0 →M′ →B→M →0, 0 →M →B′ →M′ →0, where B and B′ are uniquely determined up to isomorphism. We then have [GLS2, Theorem 2] ϕMϕM′ =ϕB +ϕB′. (28) It follows from these properties [GLS1] that categorical mutations in the category mod(Λ) (in the sense of Deﬁnition 3.9(c)) are tran”
Wall and chamber structure for finite-dimensional algebras (2018, background): “In particular, there are several representation theoretic categorifications of cluster algebras, see for instance [13, 12, 20].”
Preprojective algebras and cluster algebras (2008, background): “the Ti’s are indecomposable and pairwise non isomorphic, and d= r−rK = ℓ(w 0)− ℓ(wK 0) = ℓ(wKw 0). Assume thatthe injective summandsofTarethe lastnones, namelyTd−n+1,...,Td. Relying on the results of [14] we show in [17] the following Theorem 15.1. Let k≤ d− n. There exists a unique indecomposable m”
Derived equivalences from mutations of quivers with potential (2009, background): “ cluster algebras and those of tilting theory in the representation theory of ﬁnite-dimensional algebras. The cluster category, invented in [4], and the module category over the preprojective algebra [20] provided a conceptual framework for this remarkable phenomenon. These categories are triangulat”
Maximal modifications and Auslander–Reiten duality for non-isolated singularities (2010, methodology): “move successfully from the cluster tilting level to the maximal modiﬁcation level below, and is entirely analogous to the symmetry given by [CB, Lemma 1] as used in the work of Geiss–Leclerc–Schr¨oer [GLS]. 1.3. Maximal Modiﬁcations and NCCRs. Here we introduce our main deﬁnitions, namely modifying,”
Quiver varieties and cluster algebras (2009, background): “A closest result is the work of Geiss, Leclerc, and Schröer [25], [26], where the cluster algebra is realized as a space of constructible functions on ΛW , the space of nilpotent representations of the preprojective algebra.”
Classifying tilting complexes over preprojective algebras of Dynkin type (2015, background): “Note that these ideals are quite useful to study structure of categories [2,14,26,31], and they also play important roles in Geiss–Leclerc–Schröer’s construction of cluster monomials of certain types of cluster algebras [19,21] and Baumann–Kamnitzer–Tingley’s theory of affine MV polytopes [8].”