Cluster Categories and Their Relation to Cluster Algebras, Semi-invariants and Homology of Torsion Free Nilpotent Groups
نویسنده
چکیده
The structure of cluster categories [BMRRT] is well suited for the combinatorics of cluster algebras [FZ1] with the main correspondence being between tilting objects and clusters. Furthermore it was shown in [IOTW] that there is a close relation between domains of virtual semi-invariants and simplicial complexes associated to cluster categories. Also the same simplicial complexes associated to cluster categories are related to the Igusa-Orr pictures in the homology of nilpotent groups. 0. Introduction The purpose of this paper is to define and relate several, quite different notions, and therefore the paper is mostly a survey paper, including many results without proofs and also some indications about work in progress. With this in mind, the paper is divided in the following way. In Section 1 we define and state main properties of cluster categories. The results are mostly from [BMRRT]. In Section 2, cluster algebras with no coefficients are defined (2.1) and several results about their relation to cluster categories are given . We consider only acyclic case; in (2.2) the main bijection theorem between cluster variables and indecomposable objects of the cluster category is stated; in (2.3) the denominators of nonpolynomial cluster variables are described in terms of dimension vectors of exceptional indecomposable representations; in (2.4) weak positivity condition of cluster variables is stated as a consequence of the quite technical proposition, and the proof of the bijection theorem is given. Also, Cluster determines seed conjecture is proved in 2.5. Section 3 is mostly survey section, where we first recall definitions and theorems about classical semi-invariants, and then give definition and properties of virtual representation spaces and virtual semi-invariants. We only state the main theorems and for proofs refer to [IOTW]. Finally, Section 4 contains a short summary of the results about homology of nilpotent groups and the relation to the simplicial complex of tilting object in the corresponding cluster category.
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