### Abstract

We define the cluster algebra associated with the Q-system for the Kirillov-Reshetikhin characters of the quantum affine algebra U_{q}(g{fraktur}̂) for any simple Lie algebra, generalizing the simply-laced case treated in (Kedem in Q-systems as cluster algebras. arXiv:0712.2695 [math.RT], 2007). We describe some special properties of this cluster algebra, and explain its relation to the deformed Q-systems which appeared on our proof of the combinatorial-KR conjecture. We prove that the polynomiality of the cluster variables in terms of the "initial cluster seeds", including solutions of the Q-system, is a consequence of the Laurent phenomenon and the boundary conditions. We also define the cluster algebra associated with T-systems, or general systems which take the form of T-systems in the bipartite case. Such systems describe the recursion relations satisfied by the q-characters of Kirillov-Reshetikhin modules and also appear in the categorification picture in terms of preprojective algebras of Geiss, Leclerc and Schröer. We give a formulation of both Q-systems and generalized T-systems as cluster algebras with coefficients. This provides a proof of the polynomiality of solutions of all such "generalized T-systems" with appropriate boundary conditions.

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

Pages (from-to) | 183-216 |

Number of pages | 34 |

Journal | Letters in Mathematical Physics |

Volume | 89 |

Issue number | 3 |

DOIs | |

State | Published - Oct 2009 |

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### Keywords

- Cluster algebras
- Kirillov-Reshetikhin modules
- Q-systems

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

**Q-systems as cluster algebras II : Cartan Matrix of Finite Type and the Polynomial Property.** / Francesco, Philippe Di; Kedem, Rinat.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Q-systems as cluster algebras II

T2 - Cartan Matrix of Finite Type and the Polynomial Property

AU - Francesco, Philippe Di

AU - Kedem, Rinat

PY - 2009/10

Y1 - 2009/10

N2 - We define the cluster algebra associated with the Q-system for the Kirillov-Reshetikhin characters of the quantum affine algebra Uq(g{fraktur}̂) for any simple Lie algebra, generalizing the simply-laced case treated in (Kedem in Q-systems as cluster algebras. arXiv:0712.2695 [math.RT], 2007). We describe some special properties of this cluster algebra, and explain its relation to the deformed Q-systems which appeared on our proof of the combinatorial-KR conjecture. We prove that the polynomiality of the cluster variables in terms of the "initial cluster seeds", including solutions of the Q-system, is a consequence of the Laurent phenomenon and the boundary conditions. We also define the cluster algebra associated with T-systems, or general systems which take the form of T-systems in the bipartite case. Such systems describe the recursion relations satisfied by the q-characters of Kirillov-Reshetikhin modules and also appear in the categorification picture in terms of preprojective algebras of Geiss, Leclerc and Schröer. We give a formulation of both Q-systems and generalized T-systems as cluster algebras with coefficients. This provides a proof of the polynomiality of solutions of all such "generalized T-systems" with appropriate boundary conditions.

AB - We define the cluster algebra associated with the Q-system for the Kirillov-Reshetikhin characters of the quantum affine algebra Uq(g{fraktur}̂) for any simple Lie algebra, generalizing the simply-laced case treated in (Kedem in Q-systems as cluster algebras. arXiv:0712.2695 [math.RT], 2007). We describe some special properties of this cluster algebra, and explain its relation to the deformed Q-systems which appeared on our proof of the combinatorial-KR conjecture. We prove that the polynomiality of the cluster variables in terms of the "initial cluster seeds", including solutions of the Q-system, is a consequence of the Laurent phenomenon and the boundary conditions. We also define the cluster algebra associated with T-systems, or general systems which take the form of T-systems in the bipartite case. Such systems describe the recursion relations satisfied by the q-characters of Kirillov-Reshetikhin modules and also appear in the categorification picture in terms of preprojective algebras of Geiss, Leclerc and Schröer. We give a formulation of both Q-systems and generalized T-systems as cluster algebras with coefficients. This provides a proof of the polynomiality of solutions of all such "generalized T-systems" with appropriate boundary conditions.

KW - Cluster algebras

KW - Kirillov-Reshetikhin modules

KW - Q-systems

UR - http://www.scopus.com/inward/record.url?scp=70350392807&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=70350392807&partnerID=8YFLogxK

U2 - 10.1007/s11005-009-0354-z

DO - 10.1007/s11005-009-0354-z

M3 - Article

AN - SCOPUS:70350392807

VL - 89

SP - 183

EP - 216

JO - Letters in Mathematical Physics

JF - Letters in Mathematical Physics

SN - 0377-9017

IS - 3

ER -