A ug 1 99 5 QUANTUM W – ALGEBRAS AND ELLIPTIC ALGEBRAS

1.1. In [1] N. Reshetikhin and the second author introduced new Poisson algebras Wq(g), which are q–deformations of the classical W–algebras. The Poisson algebra Wq(g) is by definition the center of the quantized universal enveloping algebra Uq(ĝ) at the critical level, where g is the Langlands dual Lie algebra to g. It was shown in [1] that the Wakimoto realization of Uq(ĝ ) constructed in [2] provides a homomorphism from the center of Uq(ĝ ) to a Heisenberg-Poisson algebra Hq(g). This homomorphism can be viewed as a free field realization of Wq(g). When q = 1, it becomes the well-known Miura transformation [3]. In [1] explicit formulas for this free field realization were given. The structure of these formulas is the same as that of the formulas for the spectra of transfer-matrices in integrable quantum spin chains obtained by the Bethe ansatz method [4]. This is not surprising given that these spectra can actually be computed using the center of Uq(ĝ ) at the critical level and the Wakimoto realization. For the Gaudin models, which correspond to the q = 1 case, this was explained in detail in [5].