Cluster-concealed algebras
Representation Theory
2009-12-31 v1
Abstract
The cluster-tilted algebras have been introduced by Buan, Marsh and Reiten, they are the endomorphism rings of cluster-tilting objects in cluster categories; we call such an algebra cluster-concealed in case is obtained from a preprojective tilting module. For example, all representation-finite cluster-tilted algebras are cluster-concealed. If is a representation-finite cluster-tilted algebra, then the indecomposable -modules are shown to be determined by their dimension vectors. For a general cluster-tilted algebra , we are going to describe the dimension vectors of the indecomposable -modules in terms of the root system of a quadratic form. The roots may have both positive and negative coordinates and we have to take absolute values.
Cite
@article{arxiv.0912.5004,
title = {Cluster-concealed algebras},
author = {Claus Michael Ringel},
journal= {arXiv preprint arXiv:0912.5004},
year = {2009}
}