Cluster structures on simple complex lie groups and Belavin-Drinfeld classification

M. Gekhtman; M. Shapiro; A. Vainshtein

doi:10.17323/1609-4514-2012-12-2-293-312

Cluster structures on simple complex lie groups and Belavin-Drinfeld classification

M. Gekhtman, M. Shapiro, A. Vainshtein

University of Haifa

Research output: Contribution to journal › Article › peer-review

Abstract

We study natural cluster structures in the rings of regular functions on simple complex Lie groups and Poisson-Lie structures compatible with these cluster structures. According to our main conjecture, each class in the Belavin-Drinfeld classification of Poisson-Lie structures on G corresponds to a cluster structure in O(G). We prove a reduction theorem explaining how different parts of the conjecture are related to each other. The conjecture is established for SL_n, n < 5, and for any G in the case of the standard Poisson-Lie structure.

Original language	American English
Pages (from-to)	293-312
Number of pages	20
Journal	Moscow Mathematical Journal
Volume	12
Issue number	2
DOIs	https://doi.org/10.17323/1609-4514-2012-12-2-293-312
State	Published - 2012

Keywords

Belavin-Drin-feld triple
Cluster algebra
Poisson-Lie group

All Science Journal Classification (ASJC) codes

General Mathematics

Access to Document

10.17323/1609-4514-2012-12-2-293-312

Cite this

@article{4d8f466e0cd9437c8d6a3ff1ad96c209,

title = "Cluster structures on simple complex lie groups and Belavin-Drinfeld classification",

abstract = "We study natural cluster structures in the rings of regular functions on simple complex Lie groups and Poisson-Lie structures compatible with these cluster structures. According to our main conjecture, each class in the Belavin-Drinfeld classification of Poisson-Lie structures on G corresponds to a cluster structure in O(G). We prove a reduction theorem explaining how different parts of the conjecture are related to each other. The conjecture is established for SLn, n < 5, and for any G in the case of the standard Poisson-Lie structure.",

keywords = "Belavin-Drin-feld triple, Cluster algebra, Poisson-Lie group",

author = "M. Gekhtman and M. Shapiro and A. Vainshtein",

year = "2012",

doi = "10.17323/1609-4514-2012-12-2-293-312",

language = "American English",

volume = "12",

pages = "293--312",

journal = "Moscow Mathematical Journal",

issn = "1609-3321",

publisher = "Independent University of Moscow",

number = "2",

}

TY - JOUR

T1 - Cluster structures on simple complex lie groups and Belavin-Drinfeld classification

AU - Gekhtman, M.

AU - Shapiro, M.

AU - Vainshtein, A.

PY - 2012

Y1 - 2012

N2 - We study natural cluster structures in the rings of regular functions on simple complex Lie groups and Poisson-Lie structures compatible with these cluster structures. According to our main conjecture, each class in the Belavin-Drinfeld classification of Poisson-Lie structures on G corresponds to a cluster structure in O(G). We prove a reduction theorem explaining how different parts of the conjecture are related to each other. The conjecture is established for SLn, n < 5, and for any G in the case of the standard Poisson-Lie structure.

AB - We study natural cluster structures in the rings of regular functions on simple complex Lie groups and Poisson-Lie structures compatible with these cluster structures. According to our main conjecture, each class in the Belavin-Drinfeld classification of Poisson-Lie structures on G corresponds to a cluster structure in O(G). We prove a reduction theorem explaining how different parts of the conjecture are related to each other. The conjecture is established for SLn, n < 5, and for any G in the case of the standard Poisson-Lie structure.

KW - Belavin-Drin-feld triple

KW - Cluster algebra

KW - Poisson-Lie group

UR - http://www.scopus.com/inward/record.url?scp=84864080388&partnerID=8YFLogxK

U2 - 10.17323/1609-4514-2012-12-2-293-312

DO - 10.17323/1609-4514-2012-12-2-293-312

M3 - Article

SN - 1609-3321

VL - 12

SP - 293

EP - 312

JO - Moscow Mathematical Journal

JF - Moscow Mathematical Journal

IS - 2

ER -

Cluster structures on simple complex lie groups and Belavin-Drinfeld classification

Abstract

Keywords

All Science Journal Classification (ASJC) codes

Access to Document

Other files and links

Fingerprint

Cite this