The solution of the quantum $A_1$ T-system for arbitrary boundary

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.

24 pages, 18 figures

References in corpus (5)

• Q-systems as cluster algebras II: Cartan matrix of finite type and the polynomial property
• Q-systems, Heaps, Paths and Cluster Positivity
• Q-systems as cluster algebras
• Noncommutative integrability, paths and quasi-determinants
• Discrete evolution for the zero-modes of the Quantum Liouville Model

Cited by in corpus (1)

• T-systems, networks and dimers