### Abstract

Q-systems first appeared in the analysis of the Bethe equations for the XXX model and generalized Heisenberg spin chains (Kirillov and Reshetikhin 1987 Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst. Steklov. 160 211-21, 301). Such systems are known to exist for any simple Lie algebra and many other Kac-Moody algebras. We formulate the Q-system associated with any simple, simply-laced Lie algebras in the language of cluster algebras (Fomin and Zelevinsky 2002 J. Am. Math. Soc. 15 497-529), and discuss the relation of the polynomiality property of the solutions of the Q-system in the initial variables, which follows from the representation-theoretical interpretation, to the Laurent phenomenon in cluster algebras (Fomin and Zelevinsky 2002 Adv. Appl. Math. 28 119-44).

Original language | English (US) |
---|---|

Article number | 194011 |

Journal | Journal of Physics A: Mathematical and Theoretical |

Volume | 41 |

Issue number | 19 |

DOIs | |

State | Published - May 16 2008 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Statistics and Probability
- Modeling and Simulation
- Mathematical Physics
- Physics and Astronomy(all)

### Cite this

**Q-systems as cluster algebras.** / Kedem, Rinat.

Research output: Contribution to journal › Article

*Journal of Physics A: Mathematical and Theoretical*, vol. 41, no. 19, 194011. https://doi.org/10.1088/1751-8113/41/19/194011

}

TY - JOUR

T1 - Q-systems as cluster algebras

AU - Kedem, Rinat

PY - 2008/5/16

Y1 - 2008/5/16

N2 - Q-systems first appeared in the analysis of the Bethe equations for the XXX model and generalized Heisenberg spin chains (Kirillov and Reshetikhin 1987 Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst. Steklov. 160 211-21, 301). Such systems are known to exist for any simple Lie algebra and many other Kac-Moody algebras. We formulate the Q-system associated with any simple, simply-laced Lie algebras in the language of cluster algebras (Fomin and Zelevinsky 2002 J. Am. Math. Soc. 15 497-529), and discuss the relation of the polynomiality property of the solutions of the Q-system in the initial variables, which follows from the representation-theoretical interpretation, to the Laurent phenomenon in cluster algebras (Fomin and Zelevinsky 2002 Adv. Appl. Math. 28 119-44).

AB - Q-systems first appeared in the analysis of the Bethe equations for the XXX model and generalized Heisenberg spin chains (Kirillov and Reshetikhin 1987 Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst. Steklov. 160 211-21, 301). Such systems are known to exist for any simple Lie algebra and many other Kac-Moody algebras. We formulate the Q-system associated with any simple, simply-laced Lie algebras in the language of cluster algebras (Fomin and Zelevinsky 2002 J. Am. Math. Soc. 15 497-529), and discuss the relation of the polynomiality property of the solutions of the Q-system in the initial variables, which follows from the representation-theoretical interpretation, to the Laurent phenomenon in cluster algebras (Fomin and Zelevinsky 2002 Adv. Appl. Math. 28 119-44).

UR - http://www.scopus.com/inward/record.url?scp=44449087980&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=44449087980&partnerID=8YFLogxK

U2 - 10.1088/1751-8113/41/19/194011

DO - 10.1088/1751-8113/41/19/194011

M3 - Article

AN - SCOPUS:44449087980

VL - 41

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

IS - 19

M1 - 194011

ER -