The Integrals of Motion for the Deformed W-Algebra $W_{qt}(sl_N^)$ II: Proof of the commutation relations

TL;DR

This paper constructs and proves the commutation relations of two classes of integrals of motion within the deformed W-algebra $W_{qt}(sl_N^)$, extending the understanding of elliptic deformations of $W_N$ algebra.

Contribution

It explicitly constructs two classes of commuting operators and proves their algebraic relations, advancing the theory of deformed W-algebras and integrals of motion.

Findings

Construction of two classes of commuting operators

Proofs of their commutation relations

Extension to elliptic deformation of $W_N$ algebra

Abstract

We explicitly construct two classes of infinitly many commutative operators in terms of the deformed W-algebra $W_{qt}(sl_N^)$, and give proofs of the commutation relations of these operators. We call one of them local integrals of motion and the other nonlocal one, since they can be regarded as elliptic deformation of local and nonlocal integrals of motion for the $W_N$ algebra.