Quantum principal commutative subalgebra in the nilpotent part of $U_q\widehat{s\ell}_2$ and lattice KdV variables

TL;DR

This paper develops a quantum lattice framework linking KdV polynomials to functions on a homogeneous space, utilizing a quantum principal commutative subalgebra within the nilpotent part of $U_q\widehat{s\ell}_2$ to embed lattice variables.

Contribution

It introduces a quantum lattice model for KdV polynomials and constructs an action of $U_q\widehat n_+$ on these lattice variables, defining a quantum principal commutative subalgebra.

Findings

Established a quantum lattice version of Feigin and Frenkel's construction.

Embedded lattice variables in a $U_q\widehat n_+$-module coinduced from a quantum commutative subalgebra.

Identified the nilpotent part $U_q\widehat n_+$ with its coordinate algebra.

Abstract

We propose a quantum lattice version of Feigin and E. Frenkel's constructions, identifying the KdV differential polynomials with functions on a homogeneous space under the nilpotent part of $\widehat{s\ell}_2$. We construct an action of the nilpotent part $U_q\widehat n_+$ of $U_q\widehat{s\ell}_2$ on their lattice counterparts, and embed the lattice variables in a $U_q\widehat n_+$-module, coinduced from a quantum version of the principal commutative subalgebra, which is defined using the identification of $U_q\widehat n_+$ with its coordinate algebra.