TL;DR
This paper reformulates Q-systems linked to simple Lie algebras within the framework of cluster algebras, revealing connections between polynomial solutions and the Laurent phenomenon.
Contribution
It introduces a cluster algebra formulation of Q-systems for simply-laced Lie algebras, connecting representation theory with cluster algebra properties.
Findings
Q-systems are expressed as cluster algebras for simply-laced Lie algebras
Polynomiality of solutions relates to the Laurent phenomenon
Establishes a link between representation theory and cluster algebra structures
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 -system in the initial variables, which follows from the representation-theoretical interpretation, to the Laurent phenomenon in cluster algebras.
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