Algebra
Integrable clusters
[Les amas intégrables]

Présenté par : Maxim Kontsevich

Berenstein, Arkady1 ; Greenstein, Jacob2 ; Kazhdan, David3
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, Tome 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
Berenstein, Arkady 1 ; Greenstein, Jacob 2 ; Kazhdan, David 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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Berenstein, Arkady; Greenstein, Jacob; Kazhdan, David. Integrable clusters. Comptes Rendus. Mathématique, Tome 353 (2015) no. 5, pp. 387-390. doi : 10.1016/j.crma.2015.02.006. http://www.numdam.org/articles/10.1016/j.crma.2015.02.006/

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

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

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

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

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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.).