Infinitely many commuting operators for the elliptic quantum group $U_{q,p}(\hat{sl_N})$

TL;DR

This paper constructs two classes of commuting operators for the elliptic quantum group $U_{q,p}(\hat{sl_N})$, linking them to integrable models and deformations of KdV theory, with some diagonalization problems remaining open.

Contribution

It introduces two new classes of commuting operators for the elliptic quantum group, expanding understanding of their algebraic structure and connections to integrable systems.

Findings

Construction of the integral of motion ${ m f G}_m$ and boundary transfer matrix $T_B(z)$

Relation of ${ m f G}_m$ to elliptic deformation of KdV theory

Diagonalization achieved for $T_B(z)$ using free field realization

Abstract

We construct two classes of infinitely many commuting operators associated with the elliptic quantum group $U_{q,p}(\hat{sl_N})$. We call one of them the integral of motion ${\cal G}_m$, $(m \in {\mathbb N})$ and the other the boundary transfer matrix $T_B(z)$, $(z \in {\mathbb C})$. The integral of motion ${\cal G}_m$ is related to elliptic deformation of the $N$-th KdV theory. The boundary transfer matrix $T_B(z)$ is related to the boundary $U_{q,p}(\hat{sl_N})$ face model. We diagonalize the boundary transfer matrix $T_B(z)$ by using the free field realization of the elliptic quantum group, however diagonalization of the integral of motion ${\cal G}_m$ is open problem even for the simplest case $U_{q,p}(\hat{sl_2})$.