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From triangulated categories to cluster algebras. (English) Zbl 1141.18012
Let \(Q\) be a simply laced Dynkin quiver and let \(k\) be a field. In the paper under review the authors give a realisation of the cluster algebra as a Hall algebra coming from a triangulated category. This gives at once also a geometrical interpretation for the multiplication of two cluster variables.
The underlying geometrical object is the Grassmannian of a module for a quiver algebra \(kQ\). For this purpose it is proved that there is a unique way to associate to any object of the cluster category of \(Q\) a cluster variable satisfying certain naturality properties, as described by P. Caldero and F. Chapoton [Comment. Math. Helv. 81, No. 3, 595–616 (2006; Zbl 1119.16013)]. The product of the cluster variables \(X_M\) and \(X_N\) of two objects \(M\) and \(N\) of the cluster category multiplied by the Euler characteristic of the projectivisation of \(\text{Ext}^1(M,N)\) equals to the sum over all objects \(Y\) of the cluster category which occur as middle term of a triangle starting at \(M\) and ending at \(N\), or vice versa, of the cluster variable corresponding to \(Y\) multiplied with the Euler characteristic of the projectivisation of the subset of the \(\text{Ext}^1(N,M)\) having \(Y\) as middle term, plus the corresponding coefficient of \(\text{Ext}^1(M,N)\). The proof of this most astonishing result is rather complicate and involved and uses a variety of methods.
The statement has various corollaries. First, it realises the cluster algebra as a Hall algebra. Then, it settles the positivity conjecture. A third nice application is given in terms of degenerations of objects. The paper ends with four conjectures.

18E30 Derived categories, triangulated categories (MSC2010)
16G20 Representations of quivers and partially ordered sets
16E30 Homological functors on modules (Tor, Ext, etc.) in associative algebras
17B35 Universal enveloping (super)algebras
16G70 Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers
Zbl 1119.16013
Full Text: DOI arXiv
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