A geometric description of m-cluster categories

By Karin Baur, B. Marsh. 2006

89 citations recorded by the source corpus.

We show that the m-cluster category of type An 1 is equivalent to a certain geometrically-defined category of diagonals of a regular nm + 2-gon. This generalises a result of Caldero, Chapoton and Schiffler f or m = 1. The approach uses the theory of translation quivers and their corresponding mesh categories. We also introduce the notion of the mth power of a translation quiver and show how it can be used to realise the m-cluster category in terms of the cluster category.

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

View Paper: https://www.semanticscholar.org/paper/321b685e3796faa8a79e6f5c65a733b1604b4017

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Citations in the corpus

A geometric model for cluster categories of type Dn (2006, methodology): “elations. In that realization, clusters are in one-to-one correspondence with triangulations of the polygon and mutations are given by ﬂips of diagonals in the triangulation. Recently, Baur and Marsh [BM] have generalized this model to m-cluster categories of type An. In this paper, we give a geomet”
Almost complete cluster tilting objects in generalized higher cluster categories (2010, background): “A dg k-algebra A is pseudo-compact if it is endowed with a complete separated topology which is generated by two-sided dg ideals of finite codimension.”
Cluster Categories Coming from Cyclic Posets (2013, background): “These structures can also be obtained by taking mcluster categories of type An as in [1] and taking the limit as n goes to .”
Higher-dimensional cluster combinatorics and representation theory (2010, background): “ter category called OΛ for any d-representation ﬁnite algebra Λ. Note that a diﬀerent notion of higher cluster category, the d-cluster category Cd H, for H a hereditary algebra, has been studied (see [BaM] and references therein). However Cd H is not higher dimensional in the sense of this paper: Cd”
A Geometric Model for the Module Category of a Gentle Algebra (2018, background): “[4, 6, 13, 34, 41]), and of generalised cluster categories and module categories of Jacobian algebras [11].”
On a cluster category of infinite Dynkin type, and the relation to triangulations of the infinity-gon (2012, result): “See [1], [12], [19], and [20] for the type An case.”
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,…”
Cluster tilting for higher Auslander algebras (2008, background): “There are a lot of recent work on higher cluster tilting [ABST, BaM , BT, EH, HZ1, HZ2, HZ3, JH1, JH2, L, T, W, Z, ZZ].”
Mutation in triangulated categories and rigid Cohen–Macaulay modules (2006, methodology): “c and Schr¨oer [35] applied mutation to the study of rigid modules over preprojective algebras and the coordinate rings of maximal unipotent subgroups of semisimple Lie groups. We refer the reader to [21, 22, 26, 49, 51, 13, 69, 63, 62] for more recent developments in the study of cluster categories”
Coloured quiver mutation for higher cluster categories (2008, methodology): “Section 11 discusses the example of m-cluster categories of Dynkin type An, using the model developed by Baur and Marsh [ BM1 ].”