Elliptic quantum groups $E_{\tau,\eta}(sl_2)$ and quasi-Hopf algebras

Abstract

We construct an algebra morphism from the elliptic quantum group $E_{\tau,\eta}(sl_2)$ to a certain elliptic version of the ``quantum groups in higher genus'' studied by V. Rubtsov and the first author. This provides an embedding of $E_{\tau,\eta}(sl_2)$ in an algebra ``with central extension''. In particular we construct $L^{\pm}$-operators obeying a dynamical version of the Reshetikhin--Semenov-Tian-Shansky relations. To do that, we construct the factorization of a certain twist of the latter algebra, that automatically satisfies the ``twisted cocycle condition'' of O. Babelon, D. Bernard and E. Billey, and therefore provides a solution of the dynamical Yang-Baxter equation.