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Quivers with potentials and their representations. II: Applications to cluster algebras. (English) Zbl 1208.16017
This is a continuation of the study initiated by the authors in part I [Sel. Math., New Ser. 14, No. 1, 59-119 (2008; Zbl 1204.16008)], where the concepts of mutations of quivers with potentials and their decorated representations have been developed. Now they are applied to cluster algebras to prove several conjectures stated by S. Fomin and A. Zelevinsky [in Compos. Math. 143, No. 1, 112-164 (2007; Zbl 1127.16023)] concerning so called \(\mathbf g\)-vectors and \(F\)-polynomials – certain data determined via recursive relations in terms of an \(n\)-regular tree and a skew-symmetrizable integral matrix \(B\) (exchange matrix). They are important for the structure of the associated cluster algebra, [ibid.].
The main result of the present paper asserts that the conjectures 5.4, 5.5, 6.13, 7.10(1), 7.10(2) and 7.12 of Fomin and Zelevinsky [loc. cit.] hold under the assumption that the exchange matrix \(B\) is skew-symmetric. The main idea of the proof is to interprete \(F\)-polynomials in terms of suitable quiver representations. A crucial role is also played by so called \(E\)-invariant associated to decorated representations of quivers with potentials and its homological interpretation. For another approach to the conjectures see C. Fu and B. Keller [Trans. Am. Math. Soc. 362, No. 2, 859-895 (2010; Zbl 1201.18007)].

MSC:
16G20 Representations of quivers and partially ordered sets
13F60 Cluster algebras
16G10 Representations of associative Artinian rings
16S38 Rings arising from noncommutative algebraic geometry
16D90 Module categories in associative algebras
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