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

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 $p \to 0$.