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A short proof of Kontsevich’s cluster conjecture. (English) Zbl 1266.16026
Summary: We give an elementary proof of the Kontsevich conjecture that asserts that the iterations of the noncommutative rational map \(K_r\colon(x,y)\mapsto(xyx^{-1},(1+y^r)x^{-1})\) are given by noncommutative Laurent polynomials.

16S38 Rings arising from noncommutative algebraic geometry
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
13F60 Cluster algebras
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[1] Berenstein, A.; Zelevinsky, A., Quantum cluster algebras, Adv. math., 195, 2, 405-455, (2005) · Zbl 1124.20028
[2] Berenstein, A.; Fomin, S.; Zelevinsky, A., Cluster algebras III: upper and lower bounds, Duke math. J., 126, 1, 1-52, (2005) · Zbl 1135.16013
[3] Di Francesco, P.; Kedem, R., Discrete non-commutative integrability: proof of a conjecture by M. Kontsevich, Int. math. res. not., 4042-4063, (2010) · Zbl 1276.16025
[4] Fomin, S.; Zelevinsky, A., Cluster algebras I: foundations, J. amer. math. soc., 15, 497-529, (2002) · Zbl 1021.16017
[5] Fomin, S.; Zelevinsky, A., The Laurent phenomenon, Adv. in appl. math., 28, 2, 119-144, (2002) · Zbl 1012.05012
[6] Fomin, S.; Zelevinsky, A., Cluster algebras II: finite type classification, Invent. math., 154, 63-121, (2003) · Zbl 1054.17024
[7] Cohn, P.M., Free rings and their relations, (1985), Academic Press London · Zbl 0659.16001
[8] Usnich, A., Non-commutative cluster mutations, Dokl. nat. acad. sci. belarus, 53, 4, 27-29, (2009) · Zbl 1267.16012
[9] A. Usnich, Non-commutative Laurent phenomenon for two variables, preprint, arXiv:1006.1211, 2010.
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