### Abstract

This paper gives a new algebraic interpretation for the algebra generated by the quantum cluster variables of the A_{r} quantum Q-system (Di Francesco and Kedem in Int Math Res Not IMRN 10:2593–2642, 2014). We show that the algebra can be described as a quotient of the localization of the quantum algebra Uq(n[u,u-1])⊂Uq(sl^2), in the Drinfeld presentation. The generating current is made up of a subset of the cluster variables which satisfy the Q-system, which we call fundamental. The other cluster variables are given by a quantum determinant-type formula, and are polynomials in the fundamental generators. The conserved quantities of the discrete evolution (Di Francesco and Kedem in Adv Math 228(1):97–152, 2011) described by quantum Q-system generate the Cartan currents at level 0, in a non-standard polarization. The rest of the quantum affine algebra is also described in terms of cluster variables.

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

Pages (from-to) | 301-341 |

Number of pages | 41 |

Journal | Letters in Mathematical Physics |

Volume | 107 |

Issue number | 2 |

DOIs | |

State | Published - Feb 1 2017 |

### Fingerprint

### Keywords

- Discrete integrable systems
- Drinfeld algebra
- Q-systems
- Quantum determinants

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

**Quantum Q systems : from cluster algebras to quantum current algebras.** / Di Francesco, Philippe; Kedem, Rinat.

Research output: Contribution to journal › Article

*Letters in Mathematical Physics*, vol. 107, no. 2, pp. 301-341. https://doi.org/10.1007/s11005-016-0902-2

}

TY - JOUR

T1 - Quantum Q systems

T2 - from cluster algebras to quantum current algebras

AU - Di Francesco, Philippe

AU - Kedem, Rinat

PY - 2017/2/1

Y1 - 2017/2/1

N2 - This paper gives a new algebraic interpretation for the algebra generated by the quantum cluster variables of the Ar quantum Q-system (Di Francesco and Kedem in Int Math Res Not IMRN 10:2593–2642, 2014). We show that the algebra can be described as a quotient of the localization of the quantum algebra Uq(n[u,u-1])⊂Uq(sl^2), in the Drinfeld presentation. The generating current is made up of a subset of the cluster variables which satisfy the Q-system, which we call fundamental. The other cluster variables are given by a quantum determinant-type formula, and are polynomials in the fundamental generators. The conserved quantities of the discrete evolution (Di Francesco and Kedem in Adv Math 228(1):97–152, 2011) described by quantum Q-system generate the Cartan currents at level 0, in a non-standard polarization. The rest of the quantum affine algebra is also described in terms of cluster variables.

AB - This paper gives a new algebraic interpretation for the algebra generated by the quantum cluster variables of the Ar quantum Q-system (Di Francesco and Kedem in Int Math Res Not IMRN 10:2593–2642, 2014). We show that the algebra can be described as a quotient of the localization of the quantum algebra Uq(n[u,u-1])⊂Uq(sl^2), in the Drinfeld presentation. The generating current is made up of a subset of the cluster variables which satisfy the Q-system, which we call fundamental. The other cluster variables are given by a quantum determinant-type formula, and are polynomials in the fundamental generators. The conserved quantities of the discrete evolution (Di Francesco and Kedem in Adv Math 228(1):97–152, 2011) described by quantum Q-system generate the Cartan currents at level 0, in a non-standard polarization. The rest of the quantum affine algebra is also described in terms of cluster variables.

KW - Discrete integrable systems

KW - Drinfeld algebra

KW - Q-systems

KW - Quantum determinants

UR - http://www.scopus.com/inward/record.url?scp=84995395293&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84995395293&partnerID=8YFLogxK

U2 - 10.1007/s11005-016-0902-2

DO - 10.1007/s11005-016-0902-2

M3 - Article

AN - SCOPUS:84995395293

VL - 107

SP - 301

EP - 341

JO - Letters in Mathematical Physics

JF - Letters in Mathematical Physics

SN - 0377-9017

IS - 2

ER -