Q-systems as cluster algebras

Abstract

Q-systems first appeared in the analysis of the Bethe equations for the XXX-model and generalized Heisenberg spin chains. 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 g in the language of cluster algebras, 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.

16 pages, 3 figures

References in corpus (1)

• Functional Relations in Solvable Lattice Models I: Functional Relations and Representation Theory

Cited by in corpus (11)

• Cluster algebras and quantum affine algebras
• Q-systems, Heaps, Paths and Cluster Positivity
• Noncommutative integrability, paths and quasi-determinants
• QQ-system and Weyl-type transfer matrices in integrable SO(2r) spin chains
• Hyperbolic 3-manifolds and Cluster Algebras
• Q-system Cluster Algebras, Paths and Total Positivity
• Quantum Q systems: From cluster algebras to quantum current algebras
• A family of linearisable recurrences with the Laurent property
• An algebraic approach to discrete time integrability
• The solution of the quantum $A_1$ T-system for arbitrary boundary
• Linear relations for Laurent polynomials and lattice equations