Defining an m-cluster category

Abstract

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.

Generalized cluster complexes via quiver representations

“…When Γ is a Dynkin diagram with trivial valuation and Ω 0 is an alternating orientation of Γ , this compatibility degree is defined in [25]. When d = 1 and Γ is a Dynkin diagram, we recover the classical compatibility degree defined in [6,27].…”

“…it is an automorphism of D. The d-cluster category of H is defined in [19,25]: We denote by D/F d the corresponding factor category. The objects are by definition the F d -orbits of objects in D, and the morphisms are given by…”

“…d-cluster categories D/τ −1 [d], as a generalization of cluster categories, were introduced by Keller [19] and Thomas [25] for d ∈ N. They are studied by Keller and Reiten [20], Palu [1,23]; see also [3] for a geometric description of d-cluster categories of type A n . d-cluster categories are triangulated categories with Calabi-Yau dimension d + 1.…”

“…For the simply-laced Dynkin case, Thomas [25] gives a realization of generalized cluster complexes by defining the d-cluster categories.…”

Calabi–Yau triangulated categories

“…In the general case, it was introduced independently by Buan-Marsh-Reineke-Reiten-Todorov [3]. The d-cluster category was introduced in [37] and first analyzed in [57]. In general, the orbit category of a triangulated category under an autoequivalence no longer admits a structure of triangulated category.…”