Elliptic Quantum Groups $E_{\tau ,\eta } (\mathfrak{s}\mathfrak{l}_2 )$ and Quasi-Hopf Algebras

Abstract:

We construct an algebra morphism from the elliptic quantum group $E_{\tau ,\eta } (\mathfrak{s}\mathfrak{l}_2 )$ to a certain elliptic version of the “quantum loop groups in higher genus” studied by V. Rubtsov and the first author. This provides an embedding of $E_{\tau ,\eta } (\mathfrak{s}\mathfrak{l}_2 )$ in an algebra “with central extension”. In particular we construct L^±-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 quantum loop algebra, that automatically satisfies the “twisted cocycle equation” of O. Babelon, D. Bernard and E. Billey, and therefore provides a solution of the dynamical Yang–Baxter equation.