122 research outputs found
Cluster structures for 2-Calabi-Yau categories and unipotent groups
We investigate cluster tilting objects (and subcategories) in triangulated
2-Calabi-Yau categories and related categories. In particular we construct a
new class of such categories related to preprojective algebras of non Dynkin
quivers associated with elements in the Coxeter group. This class of
2-Calabi-Yau categories contains the cluster categories and the stable
categories of preprojective algebras of Dynkin graphs as special cases. For
these 2-Calabi-Yau categories we construct cluster tilting objects associated
with each reduced expression. The associated quiver is described in terms of
the reduced expression. Motivated by the theory of cluster algebras, we
formulate the notions of (weak) cluster structure and substructure, and give
several illustrations of these concepts. We give applications to cluster
algebras and subcluster algebras related to unipotent groups, both in the
Dynkin and non Dynkin case.Comment: 49 pages. For the third version the presentation is revised,
especially Chapter III replaces the old Chapter III and I
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
Derived equivalence classification of the cluster-tilted algebras of Dynkin type E
We obtain a complete derived equivalence classification of the cluster-tilted
algebras of Dynkin type E. There are 67, 416, 1574 algebras in types E6, E7 and
E8 which turn out to fall into 6, 14, 15 derived equivalence classes,
respectively. This classification can be achieved computationally and we
outline an algorithm which has been implemented to carry out this task. We also
make the classification explicit by giving standard forms for each derived
equivalence class as well as complete lists of the algebras contained in each
class; as these lists are quite long they are provided as supplementary
material to this paper. From a structural point of view the remarkable outcome
of our classification is that two cluster-tilted algebras of Dynkin type E are
derived equivalent if and only if their Cartan matrices represent equivalent
bilinear forms over the integers which in turn happens if and only if the two
algebras are connected by a sequence of "good" mutations. This is reminiscent
of the derived equivalence classification of cluster-tilted algebras of Dynkin
type A, but quite different from the situation in Dynkin type D where a
far-reaching classification has been obtained using similar methods as in the
present paper but some very subtle questions are still open.Comment: 19 pages. v4: completely rewritten version, to appear in Algebr.
Represent. Theory. v3: Main theorem strengthened by including "good"
mutations (cf. also arXiv:1001.4765). Minor editorial changes. v2: Third
author added. Major revision. All questions left open in the earlier version
by the first two authors are now settled in v2 and the derived equivalence
classification is completed. arXiv admin note: some text overlap with
arXiv:1012.466
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
Torsion pairs and rigid objects in tubes
We classify the torsion pairs in a tube category and show that they are in
bijection with maximal rigid objects in the extension of the tube category
containing the Pruefer and adic modules. We show that the annulus geometric
model for the tube category can be extended to the larger category and
interpret torsion pairs, maximal rigid objects and the bijection between them
geometrically. We also give a similar geometric description in the case of the
linear orientation of a Dynkin quiver of type A.Comment: 25 pages, 13 figures. Paper shortened. Minor errors correcte
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