Search

Let Λ be a cluster-tilted algebra of finite type over an algebraically closed field and let B be one of the associated tilted algebras. We show that the B-modules, ordered from right to left in the Auslander-Reiten quiver of Λ form a maximal forward hom-orthogonal sequence of Λ-modules whose dimension vectors form the c-vectors of a maximal green sequence for Λ. Thus we give a proof of Igusa-Todorov’s conjecture. [ABSTRACT FROM AUTHOR]

For every cluster-tilted algebra of simply-laced Dynkin type we provide a companion basis which is strong, i.e., gives the set of dimension vectors of the finitely generated indecomposable modules for the cluster-tilted algebra. This shows in particular that every companion basis of a clustertilted algebra of simply-laced Dynkin type is strong. Thus we give a proof of Parsons’s conjecture. [ABSTRACT FROM AUTHOR]

Let Λ be a row-finite higher-rank graph with no sources. We identify a maximal commutative subalgebra M inside the Kumjian-Pask algebra KPR(Λ). We also prove a generalized Cuntz-Krieger uniqueness theorem for Kumjian-Pask algebras which says that a representation of KPR(Λ) is injective if and only if it is injective on M. [ABSTRACT FROM AUTHOR]