Cluster-tilted algebras

By Aslak Bakke Buan, B. Marsh, I. Reiten. 2004

356 citations recorded by the source corpus.

We introduce a new class of algebras, which we call cluster-tilted. They are by definition the endomorphism algebras of tilting objects in a cluster category. We show that their representation theory is very close to the representation theory of hereditary algebras. As an application of this, we prove a generalised version of so-called APR-tilting.

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

View Paper: https://www.semanticscholar.org/paper/c27daa8e182183e1ac100066a6e102a45c676afc

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In Algebra and Number Theory

In Geometry and Topology

In Statistical and Nonlinear Physics

Citations in the corpus

Fomin-Zelevinsky mutation and tilting modules over Calabi-Yau algebras (2006, methodology): “A similar type of program was carried out for acyclic cluster algebras, leading to the notions of cluster categories and cluster tilted algebras [BMRRT][ BMR1 ] (see [CCS] for the An case), along with related work in [GLS].”
Representations of Algebras and Related Topics (2011, background): “gories of modules over preprojective algebras of Dynkin type, and have been studied by Geiss, Leclerc and Schr¨oer in [45], [47]. Actually most of the results on cluster-tilting theory of [24], [26], [27], [45], [47] have been proved in the more general setting of Hom-ﬁnite, triangulated, 2-Calabi-Y”
On triangulated orbit categories (2005, methodology): “The category T has Serre duality if it has both a left and a right Serre functor, or equivalently, if it has a onesided Serre functor which is an equivalence, cf. [47] [13]. The following lemma is used in [ 9 ].”
Rigid modules over preprojective algebras (2005, background): “Then T = HomΛ(T2, T1) is a tilting module over EndΛ(T1), and we have EndEndΛ(T1)(T ) ∼= EndΛ(T2) op.”
Derived equivalence classification for cluster-tilted algebras of type An (2007, methodology): “In particular, by [ BMR1 ] their representation theory can be completely understood in terms of the representation theory of the corresponding hereditary algebra H. Furthermore, their relationship to tilted algebras is well understood by [ABS1, ABS2], see also”
On Cluster Algebras Arising from Unpunctured Surfaces (2007, methodology): “... rigid object of C that corresponds to the arc . The clusters in A, and hence the triangulations of (S; M), are in bijection with the cluster-tilting objects in C. Denote by TC = fX 1; X 2; : : : ; X ng the cluster- tilting object corresponding to the triangulation T =f 1; 2; : : : ; n; n+1; : : ”
Reduction of τ-Tilting Modules and Torsion Pairs (2013, background): “Thus A is a self-injective cluster-tilted algebra of type A3, see [11, 27].”