Defining an m-cluster category

By Hugh Thomas. 2006

108 citations recorded by the source corpus.

We show that a certain orbit category considered by Keller encodes the combinatorics of the m-clusters of Fomin and Reading in a fashion similar to the way the cluster category of Buan, Marsh, Reineke, Reiten, and Todorov encodes the combinatorics of the clusters of Fomin and Zelevinsky. This allows us to give type-uniform proofs of certain results of Fomin and Reading in the simply laced cases.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/0857dabebd2908d6a65c2aa1e5cd736a1ad5f442

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Citations in the corpus

Acyclic Calabi–Yau categories (2006, methodology): “For example, if Q is a connected acylic quiver, then: (i) the cluster variables of the associated cluster algebra are in bijection with the almost positive Schur roots of the associated Kac–Moody Lie algebra (this results from [CK06] by the method of [IY06, § 7]); Received 10 September 2007,…”
On a cluster category of infinite Dynkin type, and the relation to triangulations of the infinity-gon (2012, result): “2]? (4) Do the above results generalize to ‘higher cluster categories of type A∞’? See [1], [12], [19], and [20] for the type An case.”
Mutation in triangulated categories and rigid Cohen–Macaulay modules (2006, unknown): “We refer to [19, 20, 23, 45, 47, 13, 65, 59 , 58] for more recent developments in the study of cluster categories and their generalization.”
Almost complete cluster tilting objects in generalized higher cluster categories (2010, methodology): “We write perA for the perfect derived category of A, i.e. the smallest triangulated subcategory of the derived category D(A) containing A and stable under passage to direct summands.”
Auslander–Reiten theory revisited (2008, background): “(d) An interesting example of categories with n-cluster tilting subcategories is given by (n − 1)-cluster categories ([41, 43] for n = 2, and [116, 122, 30, 119] for n ≥ 2), which are n-Calabi-Yau triangulated categories constructed from the derived categories of path algebras [92].”
Cluster categories (2010, methodology): “In the Dynkin case the concept of m-clusters has been introduced in [46], and it was shown in ([109],[115]) that the m-cluster category provides a categorification.”
A geometric description of m-cluster categories (2006, methodology): “a connection between the m-cluster category associated to kQ and the diagonals considered here was given in [ Tho ].”