81 research outputs found
Ordered Exchange Graphs
The exchange graph of a cluster algebra encodes the combinatorics of
mutations of clusters. Through the recent "categorifications" of cluster
algebras using representation theory one obtains a whole variety of exchange
graphs associated with objects such as a finite-dimensional algebra or a
differential graded algebra concentrated in non-positive degrees. These
constructions often come from variations of the concept of tilting, the
vertices of the exchange graph being torsion pairs, t-structures, silting
objects, support -tilting modules and so on. All these exchange graphs
stemming from representation theory have the additional feature that they are
the Hasse quiver of a partial order which is naturally defined for the objects.
In this sense, the exchange graphs studied in this article can be considered as
a generalization or as a completion of the poset of tilting modules which has
been studied by Happel and Unger. The goal of this article is to axiomatize the
thus obtained structure of an ordered exchange graph, to present the various
constructions of ordered exchange graphs and to relate them among each other.Comment: References updated, and Theorem A.7 adde
Estimate of the number of one-parameter families of modules over a tame algebra
The problem of classifying modules over a tame algebra A reduces to a block
matrix problem of tame type whose indecomposable canonical matrices are zero-
or one-parameter. Respectively, the set of nonisomorphic indecomposable modules
of dimension at most d divides into a finite number f(d,A) of modules and
one-parameter series of modules.
We prove that the number of m-by-n canonical parametric block matrices with a
given partition into blocks is bounded by 4^s, where s is the number of free
entries (which is at most mn), and estimate the number f(d,A).Comment: 23 page
Estimate of the number of one-parameter families of modules over a tame algebra
The problem of classifying modules over a tame algebra A reduces to a block
matrix problem of tame type whose indecomposable canonical matrices are zero-
or one-parameter. Respectively, the set of nonisomorphic indecomposable modules
of dimension at most d divides into a finite number f(d,A) of modules and
one-parameter series of modules.
We prove that the number of m-by-n canonical parametric block matrices with a
given partition into blocks is bounded by 4^s, where s is the number of free
entries (which is at most mn), and estimate the number f(d,A).Comment: 23 page
On the Cluster Category of a Marked Surface
We study in this paper the cluster category C(S,M) of a marked surface (S,M).
We explicitly describe the objects in C(S,M) as direct sums of homotopy classes
of curves in (S,M) and one-parameter families related to closed curves in
(S,M). Moreover, we describe the Auslander-Reiten structure of the category
C(S,M) in geometric terms and show that the objects without self-extensions in
C(S,M) correspond to curves in (S,M) without self-intersections. As a
consequence, we establish that every rigid indecomposable object is reachable
from an initial triangulation.Comment: 33 pages, we add a new corollary 1.6 which shows there is a bijection
between triangulations of (S,M) and the cluster-tilting objects of C(S,M),
and every rigid indecomposable object is reachable from an initial
triangulatio
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