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

Bharath Narayanan

paper

Two Equivalent Realizations of Trigonometric Dynamical Affine Quantum Group $U_{q, x} (s l_{2}) = U_{q, λ} (s l_{2})$ , Drinfeld Currents and Hopf Algebroid Structures

arXiv:1405.7833

Abstract

Two new realizations, denoted $U_{q, x} (g l_{2})$ and $U (R_{q, x} (g l_{2}))$ of the trigonometric dynamical quantum affine algebra $U_{q, λ} (g l_{2})$ are proposed, based on Drinfeld-currents and $R LL$ relations respectively, along with a Heisenberg algebra ${P, Q}$ , with $x = q^{2 P}$ . Here $P$ plays the role of the dynamical variable $λ$ and $Q = \frac{\partial}{\partial P}$ . An explicit isomorphism from $U_{q, x} (g l_{2})$ to $U (R_{q, x} (g l_{2}))$ is established, which is a dynamical extension of the Ding-Frenkel isomorphism of $U_{q} (g l_{2})$ with $U (R_{q} (g l_{2}))$ 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 $U_{q, x} (s l_{2})$ is isomorphic to $U (R_{q, x} (s l_{2}))$ . The dynamical construction is based on the degeneration of the elliptic quantum algebra $U_{q, p} (s l_{2})$ of Jimbo, Konno et al. as the elliptic variable $p \to 0$ .

37 pages, 2 figures. Misprints corrected and improvements of v1

Two Equivalent Realizations of Trigonometric Dynamical Affine Quantum Group Uq,x​(sl2​​)=Uq,λ​(sl2​​), Drinfeld Currents and Hopf Algebroid Structures

Abstract

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Two Equivalent Realizations of Trigonometric Dynamical Affine Quantum Group $U_{q, x} (s l_{2}) = U_{q, λ} (s l_{2})$ , Drinfeld Currents and Hopf Algebroid Structures