Quantum Currents Realization Of

We review the construction by G. Felder and the author of the realization of the elliptic quantum groups by quantum currents. . The elliptic quantum groups were introduced by G. Felder in [14]. These are algebraic objects based on a solution R(z, λ) of the dynamical Yang-Baxter equation. Here dynamical means that in addition to the spectral parameter z, the R-matrix depends on a parameter λ, which belongs to a product of elliptic curves, and that these parameters undergo shifts in the various terms of the equation. The aim of this paper is to review the construction by G. Felder and the author ([9]) of the realization of elliptic quantum groups Eτ,η(sl2) by quantum current algebras. This construction relies on quasi-Hopf algebra techniques. We introduce (sect. 2) a quantum loop algebra U~g(τ) (τ is the elliptic parameter, g = sl2) that presents analogies with Eτ,η(sl2). Namely, it has the property that the image in representations of its classical r-matrix coincides with the classical limit of R(z, λ). U~g(τ) is endowed with “Drinfeld-type” coproducts ∆ and ∆̄ (see [6]), conjugated by a twist F (sect. 2.3). Then our goal is to construct, in this algebra, a solution of the DYBE yielding R(z, λ) in finite-dimensional representations. For that, we make use of a result of O. Babelon, D. Bernard and E. Billey (BBB). This result extends to the dynamical situation the theory of Drinfeld twists (see [7]): it states that a solution of the so-called twisted Hopf cocycle equation, in some quasi-triangular Hopf algebra, yields a solution of the DYBE at the algebra level. To construct such a solution, we solve a factorization problem for the twist F (sect. 4.3). This factorization in turn relies on some results on Hopf algebra pairings within the quantum loop algebra. After we have obtained a DYBE solution in U~g(τ) , we study its representations and construct from it L-operators, that satisfy RLL relations which are exactly the elliptic quantum groups relations (sect. 4.6). Date: August 1997. 1