1,861 research outputs found
A geometric description of the m-cluster categories of type D_n
We show that the m-cluster category of type D_n is equivalent to a certain
geometrically-defined category of arcs in a punctured regular nm-m+1-gon. This
generalises a result of Schiffler for m=1. We use the notion of the mth power
of a translation quiver to realise the m-cluster category in terms of the
cluster category.Comment: 14 pages, 11 figure
Coloured quivers for rigid objects and partial triangulations: The unpunctured case
We associate a coloured quiver to a rigid object in a Hom-finite
2-Calabi--Yau triangulated category and to a partial triangulation on a marked
(unpunctured) Riemann surface. We show that, in the case where the category is
the generalised cluster category associated to a surface, the coloured quivers
coincide. We also show that compatible notions of mutation can be defined and
give an explicit description in the case of a disk. A partial description is
given in the general 2-Calabi-Yau case. We show further that Iyama-Yoshino
reduction can be interpreted as cutting along an arc in the surface.Comment: 29 pages, 17 figures. Discussion in Section 6 clarified and expanded.
Some minor corrections, clarification of notatio
From triangulated categories to module categories via localisation II: Calculus of fractions
We show that the quotient of a Hom-finite triangulated category C by the
kernel of the functor Hom(T, -), where T is a rigid object, is preabelian. We
further show that the class of regular morphisms in the quotient admit a
calculus of left and right fractions. It follows that the Gabriel-Zisman
localisation of the quotient at the class of regular morphisms is abelian. We
show that it is equivalent to the category of finite dimensional modules over
the endomorphism algebra of T in C.Comment: 21 pages; no separate figures. Minor changes. To appear in Journal of
the London Mathematical Society (published version is different
Denominators of cluster variables
Associated to any acyclic cluster algebra is a corresponding triangulated
category known as the cluster category. It is known that there is a one-to-one
correspondence between cluster variables in the cluster algebra and exceptional
indecomposable objects in the cluster category inducing a correspondence
between clusters and cluster-tilting objects.
Fix a cluster-tilting object T and a corresponding initial cluster. By the
Laurent phenomenon, every cluster variable can be written as a Laurent
polynomial in the initial cluster. We give conditions on T equivalent to the
fact that the denominator in the reduced form for every cluster variable in the
cluster algebra has exponents given by the dimension vector of the
corresponding module over the endomorphism algebra of T.Comment: 22 pages; one figur
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