Comptes Rendus
no. 5
Algebra
Integrable clusters
[Les amas intégrables]

Présenté par : Maxim Kontsevich

Arkady Berenstein1 ; Jacob Greenstein2 ; David Kazhdan3
1 Department of Mathematics, University of Oregon, Eugene, OR 97403, USA
2 Department of Mathematics, University of California, Riverside, CA 92521, USA
3 Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel
Comptes Rendus. Mathématique, Volume 353 (2015) no. 5, pp. 387-390.

Le but de cet article est d'étudier les amas quantiques dont les variables d'amas (mais pas les coefficients) commutent entre elles. Cette propriété est préservée par les mutations si l'on commence par une graine quantique principale. Remarquablement, elle est équivalente à la conjecture notoire sur la cohérence de signes qui a été récemment démontrée par M. Gross, P. Hacking, S. Keel et M. Kontsevich.

The goal of this note is to study quantum clusters in which cluster variables (not coefficients) commute which each other. It turns out that this property is preserved by mutations if one starts with a principal quantum seed. Remarkably, this is equivalent to the celebrated sign coherence conjecture recently proved by M. Gross, P. Hacking, S. Keel, and M. Kontsevich.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2015.02.006
Arkady Berenstein 1 ; Jacob Greenstein 2 ; David Kazhdan 3

1 Department of Mathematics, University of Oregon, Eugene, OR 97403, USA
2 Department of Mathematics, University of California, Riverside, CA 92521, USA
3 Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel 
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Arkady Berenstein; Jacob Greenstein; David Kazhdan. Integrable clusters. Comptes Rendus. Mathématique, Volume 353 (2015) no. 5, pp. 387-390. doi : 10.1016/j.crma.2015.02.006. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.02.006/

[1] Vladimir Arnol'd Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, vol. 60, Springer-Verlag, 1989

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

[3] Arkady Berenstein; Andrei Zelevinsky Triangular bases in quantum cluster algebras, Int. Math. Res. Not., Volume 2014 (2014) no. 6, pp. 1651-1688

[4] Sergey Fomin; Andrei Zelevinsky Cluster algebras. I. Foundations, J. Amer. Math. Soc., Volume 15 (2002) no. 2, pp. 497-529

[5] Sergey Fomin; Andrei Zelevinsky Cluster algebras, IV. Coefficients, Compos. Math., Volume 143 (2007) no. 1, pp. 112-164

[6] Mark Gross; Paul Hacking; Sean Keel; Maxim Kontsevich Canonical bases for cluster algebras | arXiv

[7] Tomoki Nakanishi; Andrei Zelevinsky On tropical dualities in cluster algebras, Algebraic Groups and Quantum Groups, Contemp. Math., vol. 565, Amer. Math. Soc., Providence, RI, 2012, pp. 217-226

[8] Nicolai Reshetikhin; Milen Yakimov Deformation quantization of Lagrangian fiber bundles, 1999, Dijon, France (Math. Phys. Stud.), Volume vol. 22, Kluwer Academic Publishers, Dordrecht, The Netherlands (2000), pp. 263-287 MR1805921 (2002h:53156)

Cité par Sources :

☆ The authors were partially supported by the BSF grant no. 2012365 (A. B. and D. K.), NSF grant DMS-1403527 (A. B.), the ERC grant no. 247049 (D. K.) and the Simons Foundation collaboration grant no. 245735 (J. G.).

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