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.