Two Equivalent Realizations of Trigonometric Dynamical Affine Quantum Group $U_{q,x}(\widehat{sl_2})=U_{q,\lambda}(\widehat{sl_2})$, Drinfeld Currents and Hopf Algebroid Structures

TL;DR

This paper introduces two new realizations of the trigonometric dynamical quantum affine algebra $U_{q, ext{lambda}}(\\widehat{gl_2})$, establishes an explicit isomorphism between them, and explores their Hopf algebroid structures, extending known quantum algebra isomorphisms.

Contribution

It presents two equivalent realizations of the dynamical quantum affine algebra and constructs an explicit isomorphism, extending the Ding-Frenkel isomorphism to the dynamical setting.

Findings

Explicit isomorphism between Drinfeld currents and RLL realizations.

Introduction of Hopf algebroid structures and a dynamical determinant.

Extension of the Ding-Frenkel isomorphism to the dynamical case.

Abstract

Two new realizations, denoted $U_{q,x}(\widehat{gl_2})$ and $U(R_{q,x}(\widehat{gl_2}))$ of the trigonometric dynamical quantum affine algebra $U_{q,\lambda}(\widehat{gl_2})$ are proposed, based on Drinfeld-currents and $RLL$ relations respectively, along with a Heisenberg algebra $\left\{P,Q\right\}$, with $x=q^{2P}$. Here $P$ plays the role of the dynamical variable $\lambda$ and $Q=\frac{\partial}{\partial P}$. An explicit isomorphism from $U_{q,x}(\widehat{gl_2})$ to $U(R_{q,x}(\widehat{gl_2}))$ is established, which is a dynamical extension of the Ding-Frenkel isomorphism of $U_{q}(\widehat{gl_2})$ with $U(R_{q}(\widehat{gl_2}))$ between the Drinfeld realization and the Reshetikhin-Tian-Shanksy construction of quantum affine algebras. Hopf algebroid structures and an affine dynamical determinant element are introduced and it is shown that $U_{q,x}(\widehat{sl_2})$ is isomorphic to $U(R_{q,x}(\widehat{sl_2}))$. The dynamical construction is based on the degeneration of the elliptic quantum algebra $U_{q,p}(\widehat{sl_2})$ of Jimbo, Konno et al. as the elliptic variable $p \to 0$.