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Quivers with potentials and their representations II

Quivers with potentials and their representations II,10.1090/S0894-0347-10-00662-4,Journal of The American Mathematical Society,Harm Derksen,Jerzy Wey

Quivers with potentials and their representations II   (Citations: 39)
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We continue the study of quivers with potentials and their representations initiated in the first paper of the series. Here we develop some applications of this theory to cluster algebras. As shown in the ``Cluster algebras IV'' paper, the cluster algebra structure is to a large extent controlled by a family of integer vectors called {g} -vectors, and a family of integer polynomials called F -polynomials. In the case of skew-symmetric exchange matrices we find an interpretation of these {g} -vectors and F -polynomials in terms of (decorated) representations of quivers with potentials. Using this interpretation, we prove most of the conjectures about {g} -vectors and F -polynomials made in loc. cit.
Journal: Journal of The American Mathematical Society - J AMER MATH SOC , vol. 23, no. 3, pp. 749-790, 2010
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    • ...This allows us to establish the link to previous approaches, especially to the one based on the cluster category and the one based on decorated representations [36] [12] [11]...
    • ...This was previously proved in [3] and [11]...

    Bernhard Kelleret al. Derived equivalences from mutations of quivers with potential

    • ...More recently, Derksen et al. [18, 19] have obtained a far-reaching generalization of [53]...

    Laurent Demonet. Categorification of Skew-symmetrizable Cluster Algebras

    • ...This conjecture was proven in [3 ]f or the case when B 0 is skew-symmetric, and also in [9] in the situation where the cluster algebra admits a certain categorification...
    • ...The next two propositions are an immediate consequence of Proposition 5.7 and Corollary 5.8 of [3]:...
    • ...This formula was given in [3] in terms of representations of quivers with potentials...
    • ...We recall some notation and definitions from [3], omitting certain technical details which are not needed here...
    • ...A certain linear map γk : Mout(k) → Min(k) was defined in [3]...
    • ...Proof The proposition follows immediately from Theorem 5.1 and Proposition 5.7 of [3]...

    Thao Tran. F-Polynomials in Quantum Cluster Algebras

    • ...For their studies the Euler characteristics of a class of projective varieties, called quiver Grassmannians, are important (see [5, 10])...

    Nicolas Haupt. Euler Characteristics of Quiver Grassmannians and Ringel-Hall Algebras...

    • ...Though mainly combinatorial in their conception, many important questions about cluster algebras were answered by introducing some extra structure like for example a “categorification” [9,14 ]o r some Poisson geometric context [19]...

    Michael Barotet al. Tubular cluster algebras I: categorification

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