Cluster-tilted algebras of finite representation type

By Aslak Bakke Buan, Robert J. Marsh, I. Reiten. 2005

114 citations recorded by the source corpus.

Abstract We investigate the cluster-tilted algebras of finite representation type over an algebraically closed field. We give an explicit description of the relations for the quivers for finite representation type. As a consequence we show that a (basic) cluster-tilted algebra of finite type is uniquely determined by its quiver. Also some necessary conditions on the shapes of quivers of cluster-tilted algebras of finite representation type are obtained along the way.

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

View Paper: https://www.semanticscholar.org/paper/b4f33ea912901782da204418419a4a35b6561152

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In Algebra and Number Theory

In Geometry and Topology

In Statistical and Nonlinear Physics

Citations in the corpus

Acyclic Calabi–Yau categories (2006, background): “If k is a field and Q a quiver without oriented cycles, the associated cluster category CQ is the ‘largest’ 2-Calabi–Yau category under the derived category of representations of Q over k.”
Cluster categories (2010, background): “In the case of finite representation type this was proved in [21], where also the relations were described.”
Tilting theory and cluster combinatorics (2004, methodology): “2 has been solved in [BMR05], and partially in [CCS04b].”
Cluster‐tilted algebras as trivial extensions (2006, background): “Since then, these algebras have been the subject of many investigations, see, for instance, [7, 8, 9, 10, 11, 12, 13, 21].”
Derived equivalence classification for cluster-tilted algebras of type An (2007, methodology): “In this section we give the relations on the quivers of cluster-tilted algebras of type An, which are given in [CCS1] and have been generalised in [CCS2] and [ BMR3 ].”
Lattice Properties of Oriented Exchange Graphs and Torsion Classes (2015, background): “Cluster-tilted algebras of Dynkin type can be described explicitly as bound quiver algebras (see [6]).”
Almost complete cluster tilting objects in generalized higher cluster categories (2010, unknown): “R T ] 9 J an 2 01 2”
Cluster-tilted algebras are Gorenstein and stably Calabi–Yau (2005, methodology): “We refer to [23] for more background on cluster algebras and to [1] [2] [ 12 ] [13] [14] [15] [16] [18] [19] [28] [33] [34] [46] [47] for some recent developments in the study of their links with representation theory.”
Cluster-tilted algebras and slices (2007, background): “These algebras have been studied by several authors, see, for instance, [1,8,9,10,12,17].”