A short proof of Kontsevich's cluster conjecture

Arkady Berenstein; Vladimir Retakh

doi:10.1016/j.crma.2011.01.004

Combinatorics/Algebra

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

Présenté par : Maxime Kontsevich

Arkady Berenstein ¹ ; Vladimir Retakh ²

¹ Department of Mathematics, University of Oregon, Eugene, OR 97403, USA
² Department of Mathematics, Rutgers University, Piscataway, NJ 08854, USA

Comptes Rendus. Mathématique, Volume 349 (2011) no. 3-4, pp. 119-122.

Résumé
Abstract

Nous proposons une démonstration élémentaire d'une conjoncture de Kontsevich qui affirme que l'itération de l'application non-commutative rationnelle $K_{r} : (x, y) \mapsto (x y x^{- 1}, (1 + y^{r}) x^{- 1})$ est donnée par des polynômes de Laurent non-commutatifs.

We give an elementary proof of the Kontsevich conjecture that asserts that the iterations of the noncommutative rational map $K_{r} : (x, y) \mapsto (x y x^{- 1}, (1 + y^{r}) x^{- 1})$ are given by noncommutative Laurent polynomials.

Reçu le : 2010-11-18
Accepté le : 2011-01-04
Publié le : 2011-01-20

DOI : 10.1016/j.crma.2011.01.004

Affiliations des auteurs :

Arkady Berenstein ¹ ; Vladimir Retakh 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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     title = {A short proof of {Kontsevich's} cluster conjecture},
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     pages = {119--122},
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     year = {2011},
     doi = {10.1016/j.crma.2011.01.004},
     language = {en},
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Arkady Berenstein; Vladimir Retakh. A short proof of Kontsevich's cluster conjecture. Comptes Rendus. Mathématique, Volume 349 (2011) no. 3-4, pp. 119-122. doi : 10.1016/j.crma.2011.01.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.01.004/

Bibliographie
Cité par

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

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

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

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

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

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

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

[8] A. Usnich 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

Cité par Sources :

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

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