Generalized cluster complexes via quiver representations

By Bin Zhu. 2006

77 citations recorded by the source corpus.

Abstract We give a quiver representation theoretic interpretation of generalized cluster complexes defined by Fomin and Reading. Using d-cluster categories defined by Keller as triangulated orbit categories of (bounded) derived categories of representations of valued quivers, we define a d-compatibility degree (−∥−) on any pair of “colored” almost positive real Schur roots which generalizes previous definitions on the noncolored case and call two such roots compatible, provided that their d-compatibility degree is zero. Associated to the root system Φ corresponding to the valued quiver, using this compatibility relation, we define a simplicial complex which has colored almost positive real Schur roots as vertices and d-compatible subsets as simplices. If the valued quiver is an alternating quiver of a Dynkin diagram, then this complex is the generalized cluster complex defined by Fomin and Reading.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8af16c17d57531224da6b8769b2a4f69fa68e8cf

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Geometry and Topology

In Mathematical Physics

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,…”
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.”
Coloured quiver mutation for higher cluster categories (2008, methodology): “Hence there is map T (e+1) i → Tk ∐ Z such that Tj ∐ X ′ → Tk ∐ Z = (Tj ∐ X ′ →”
A geometric description of m-cluster categories (2006, methodology): “alence τ−1Sm. We thus obtain a geometric description of this category in terms of Π. The m-cluster category Db(kQ)/τ−1Sm associated to kQwas introduced in [Kel] and has also been studied in [Tho] and [Zhu]. It is a generalisation of the cluster category deﬁned in [CCS1] (for type A) and [BMRRT] (gen”
Cluster tilting for higher Auslander algebras (2008, methodology): “For the case n = 1, we have C[ Z ] = D, which means the well-known fact that any object in D is a direct sum of stalk complexes if � is hereditary.”
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].”
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.”