Graded mutation in cluster categories coming from hereditary categories with a tilting object

Abstract

a b s t r a c tWe present a graded mutation rule for quivers of cluster-tilted algebras. Furthermore, we give a technique for recovering a cluster-tilting object from its graded quiver in the cluster category of coh X.

Tubular cluster algebras II: Exponential growth

“…Note that the endomorphism ring of a cluster tilting object in C is in general not determined by its quiver. For example, this occurs for the tubular cluster category of weight type (2, 2, 2, 2), see [5,Expl. 6.12].…”

Mutating loops and 2-cycles in 2-CY triangulated categories

“…For instance, let H be a hereditary abelian k-category over a field k, with finite dimensional Hom-spaces and Ext 1 -spaces having a tilting object T . If H has no nonzero projective (or nonzero injective) objects, we know that every almost complete tilting object (that is, a tilting object where one indecomposable summand is removed) has exactly two complements (see [H2,HU,BOW2]). Thus we can always replace any indecomposable summand of T , to obtain a new tilting object T .…”