Quantum Toroidal Comodule Algebra of Type $A_{n-1}$ and Integrals of Motion

TL;DR

This paper introduces a new algebraic structure, $\

Contribution

It generalizes the algebra $\

Findings

Existence of a family of commutative subalgebras

Provides a uniform description of deformed W algebras for Lie (super)algebras of types BCD

Extends the structure from the rank 1 case to higher ranks

Abstract

We introduce an algebra $\mathcal{K}_n$ which has a structure of a left comodule over the quantum toroidal algebra of type $A_{n-1}$. Algebra $\mathcal{K}_n$ is a higher rank generalization of $\mathcal{K}_1$, which provides a uniform description of deformed $W$ algebras associated with Lie (super)algebras of types BCD. We show that $\mathcal{K}_n$ possesses a family of commutative subalgebras.