Geodesic
Combinatorics/Algebra
A short proof of Kontsevich's cluster conjecture
[Une courte démonstration d'une conjoncture de Kontsevich]

Présenté par : Maxime Kontsevich

Berenstein, Arkady1 ; Retakh, Vladimir2
1 Department of Mathematics, University of Oregon, Eugene, OR 97403, USA
2 Department of Mathematics, Rutgers University, Piscataway, NJ 08854, USA
Comptes Rendus. Mathématique, Tome 349 (2011) no. 3-4, pp. 119-122.

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We give an elementary proof of the Kontsevich conjecture that asserts that the iterations of the noncommutative rational map Kr:(x,y)↦(xyx−1,(1+yr)x−1) are given by noncommutative Laurent polynomials.

Nous proposons une démonstration élémentaire d'une conjoncture de Kontsevich qui affirme que l'itération de l'application non-commutative rationnelle Kr:(x,y)↦(xyx−1,(1+yr)x−1) est donnée par des polynômes de Laurent non-commutatifs.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2011.01.004

Berenstein, Arkady 1 ; Retakh, Vladimir 2

1 Department of Mathematics, University of Oregon, Eugene, OR 97403, USA
2 Department of Mathematics, Rutgers University, Piscataway, NJ 08854, USA 
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Berenstein, Arkady; Retakh, Vladimir. A short proof of Kontsevich's cluster conjecture. Comptes Rendus. Mathématique, Tome 349 (2011) no. 3-4, pp. 119-122. doi : 10.1016/j.crma.2011.01.004. http://geodesic.mathdoc.fr/articles/10.1016/j.crma.2011.01.004/

[1] Berenstein, A.; Zelevinsky, A. Quantum cluster algebras, Adv. Math., Volume 195 (2005) no. 2, pp. 405-455

[2] Berenstein, A.; Fomin, S.; Zelevinsky, A. Cluster algebras III: Upper and lower bounds, Duke Math. J., Volume 126 (2005) no. 1, pp. 1-52

[3] Di Francesco, P.; Kedem, R. Discrete non-commutative integrability: Proof of a conjecture by M. Kontsevich, Int. Math. Res. Not. (2010), pp. 4042-4063

[4] Fomin, S.; Zelevinsky, A. Cluster algebras I: Foundations, J. Amer. Math. Soc., Volume 15 (2002), pp. 497-529

[5] Fomin, S.; Zelevinsky, A. The Laurent phenomenon, Adv. in Appl. Math., Volume 28 (2002) no. 2, pp. 119-144

[6] Fomin, S.; Zelevinsky, A. Cluster algebras II: Finite type classification, Invent. Math., Volume 154 (2003), pp. 63-121

[7] Cohn, P.M. Free Rings and Their Relations, Academic Press, London, 1985

[8] Usnich, A. Non-commutative cluster mutations, Dokl. Nat. Acad. Sci. Belarus, Volume 53 (2009) no. 4, pp. 27-29

[9] A. Usnich, Non-commutative Laurent phenomenon for two variables, preprint, , 2010. | arXiv

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☆ The authors were supported in part by the NSF grant DMS #0800247.