Quantum Q systems: From cluster algebras to quantum current algebras

TL;DR

This paper explores the relationship between quantum Q-systems, cluster algebras, and quantum current algebras, revealing new interpretations of conserved quantities and algebraic structures within the framework of Lie algebra representations.

Contribution

It demonstrates that the quantum Q-system for $A_r$ can be viewed as a quotient of the quantum current algebra in the Drinfeld presentation, linking different algebraic frameworks.

Findings

Quantum Q-system is a quotient of quantum current algebra.

Conserved quantities are interpreted via Cartan currents at level 0.

Quantum cluster algebra generators relate to the current algebra in a non-standard polarization.

Abstract

In this paper, we recall our renormalized quantum Q-system associated with representations of the Lie algebra $A_r$, and show that it can be viewed as a quotient of the quantum current algebra $U_q({\mathfrak n}[u,u^{-1}])\subset U_q(\widehat{\mathfrak sl}_2)$ in the Drinfeld presentation. Moreover, we find the interpretation of the conserved quantities in terms of Cartan currents at level 0, and the rest of the current algebra, in a non-standard polarization in terms of generators in the quantum cluster algebra.