Quantum cluster variables via Serre polynomials. (English summary) With an appendix by Bernhard Keller. J. Reine Angew. Math. 668 (2012), 149–190.1435-5345 Cluster algebras were discovered by S. Fomin and A. Zelevinsky [J. Amer. Math. Soc. 15 (2002), no. 2, 497–529; MR1887642 (2003f:16050)]. A skew-symmetric cluster algebra A is a subalgebra of the rational function field Q(x1,..., xn) of n indeterminates equipped with a distinguished set of variables (cluster variables) grouped into overlapping subsets (clusters) consisting of n elements, defined by a recursive procedure (mutation) on quivers. A. Berenstein and Zelevinsky [Adv. Math. 195 (2005), no. 2, 405–455; MR2146350 (2006a:20092)] introduced the quantum version of cluster algebras. It is shown that many results on cluster algebras (the Laurent phenomenon, the classification of finite-type cluster algebras) extend to the quantum case. The important open problem is to show that quantum cluster monomials have non-negative coefficients. The paper under review deals with skew-symmetric acyclic quantum cluster algebras. The quantum F-polynomials and the quantum cluster monomials are expressed in terms of Serre polynomials of quiver Grassmannians of rigid modules. It follows that the positivity conjectur
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