The Integrals of Motion for the Deformed W-Algebra $${W_{q,t}(\widehat{gl_N})}$$ . II. Proof of the Commutation Relations

Abstract

We explicitly construct two classes of infinitely many commutative operators in terms of the deformed W-algebra \({W_{q,t}(\widehat{gl_N})}\), and give proofs of the commutation relations of these operators. We call one of them local integrals of motion and the other nonlocal, since they can be regarded as elliptic deformations of local and nonlocal integrals of motion for the Virasoro algebra and the W 3 algebra [1,2].