Cluster structures for 2-Calabi–Yau categories and unipotent groups

By Aslak Bakke Buan, Osamu Iyama, I. Reiten, J. Scott. Compositio Mathematica, 2007

326 citations recorded by the source corpus.

Abstract We investigate cluster-tilting objects (and subcategories) in triangulated 2-Calabi–Yau and related categories. In particular, we construct a new class of such categories related to preprojective algebras of non-Dynkin quivers associated with elements in the Coxeter group. This class of 2-Calabi–Yau categories contains, as special cases, the cluster categories and the stable categories of preprojective algebras of Dynkin graphs. For these 2-Calabi–Yau categories, we construct cluster-tilting objects associated with each reduced expression. The associated quiver is described in terms of the reduced expression. Motivated by the theory of cluster algebras, we formulate the notions of (weak) cluster structure and substructure, and give several illustrations of these concepts. We discuss connections with cluster algebras and subcluster algebras related to unipotent groups, in both the Dynkin and non-Dynkin cases.

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/0408051520300695c01e5dba4f6827391265ed76

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Citations in the corpus

Classifying $$\tau $$τ-tilting modules over preprojective algebras of Dynkin type (2013, background): “In [14] (see also [26]), the authors studied preprojective algebras of non-Dynkin quivers via tilting theory.”
Cluster structures on strata of flag varieties (2014, methodology): “e categoriﬁcation ap proach of §1.4, which only deals with (localizations of) subrings of C[N]. In Section 3 we introduce the subcategory Cv,w and show that it has a cluster structure in the sense of [BIRS]. We also give a general recipe for constructing cluster-tilting objects ofCv,w, and we descri”
Representations of Algebras and Related Topics (2011, background): “n cluster-tilting theory of [24], [26], [27], [45], [47] have been proved in the more general setting of Hom-ﬁnite, triangulated, 2-Calabi-Yau categories with cluster-tilting objects (see [70], [63], [22]). Since then, other categories with the above properties have been constructed and investigated”
Affine Mirković-Vilonen polytopes (2011, methodology): “One of these tools is Buan, Iyama, Reiten and Scott’s tilting ideals for Λ [14].”
Acyclic Calabi–Yau categories (2006, background): “If k is algebraically closed and, for each vertex x of Q, only finitely many paths start in x and only finitely many paths end in x, then C is triangle-equivalent to the cluster category CQ. Note that under the assumptions of the corollary, the projective (right) kQ-module kQ(?, x) and the injective”
Cluster algebra structures and semicanonical bases for unipotent groups (2007, methodology): “ore we may deﬁne Cw := CM, and we have associated to every adaptable element w∈W a subcategory Cw of nil(Λ). This subcategory is, up to duality, the same as the one introduced in a diﬀerent manner in [BIRS]. To check this, one only needs to compare our maximal rigid module T∨ M (as described in Prop”
Silting mutation in triangulated categories (2010, background): “Their beautiful theory of ‘tilting mutation’ has a lot of important applications, especially in the study of cluster categories and preprojective algebras [10, 11, 13, 22, 40].”
Preprojective algebras and MV polytopes (2010, unknown): “, see for instance [6].”
Classifying tilting complexes over preprojective algebras of Dynkin type (2015, background): “In [14] (see also [26]), the authors studied preprojective algebras of non-Dynkin quivers via tilting theory.”