### Abstract

We solve the quantum version of the A _{1} T-system by use of quantum networks. The system is interpreted as a particular set of mutations of a suitable (infinite-rank) quantum cluster algebra, and Laurent positivity follows from our solution. As an application we re-derive the corresponding quantum network solution to the quantum A _{1} Q-system and generalize it to the fully non-commutative case. We give the relation between the quantum T-system and the quantum lattice Liouville equation, which is the quantized Y-system.

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

Pages (from-to) | 329-350 |

Number of pages | 22 |

Journal | Communications in Mathematical Physics |

Volume | 313 |

Issue number | 2 |

DOIs | |

State | Published - Jul 1 2012 |

### Fingerprint

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

**The Solution of the Quantum A _{1} T-System for Arbitrary Boundary.** / Di Francesco, Philippe; Kedem, Rinat.

Research output: Contribution to journal › Article

_{1}T-System for Arbitrary Boundary',

*Communications in Mathematical Physics*, vol. 313, no. 2, pp. 329-350. https://doi.org/10.1007/s00220-012-1488-x

}

TY - JOUR

T1 - The Solution of the Quantum A 1 T-System for Arbitrary Boundary

AU - Di Francesco, Philippe

AU - Kedem, Rinat

PY - 2012/7/1

Y1 - 2012/7/1

N2 - We solve the quantum version of the A 1 T-system by use of quantum networks. The system is interpreted as a particular set of mutations of a suitable (infinite-rank) quantum cluster algebra, and Laurent positivity follows from our solution. As an application we re-derive the corresponding quantum network solution to the quantum A 1 Q-system and generalize it to the fully non-commutative case. We give the relation between the quantum T-system and the quantum lattice Liouville equation, which is the quantized Y-system.

AB - We solve the quantum version of the A 1 T-system by use of quantum networks. The system is interpreted as a particular set of mutations of a suitable (infinite-rank) quantum cluster algebra, and Laurent positivity follows from our solution. As an application we re-derive the corresponding quantum network solution to the quantum A 1 Q-system and generalize it to the fully non-commutative case. We give the relation between the quantum T-system and the quantum lattice Liouville equation, which is the quantized Y-system.

UR - http://www.scopus.com/inward/record.url?scp=84862886114&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84862886114&partnerID=8YFLogxK

U2 - 10.1007/s00220-012-1488-x

DO - 10.1007/s00220-012-1488-x

M3 - Article

AN - SCOPUS:84862886114

VL - 313

SP - 329

EP - 350

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 2

ER -