FROM QUANTUM TO ELLIPTIC ALGEBRAS

It is shown that the elliptic algebra A(q,p)(<(sl)over cap>(2)(c)) at the critical level c = -2 has a multidimensional center containing som e trace-like operators t(z). A family of Poisson structures indexed by a non-negative integer and containing the q-deformed Virasoro algebra is constructed on this center. We show also that t(z) close an exchan ge algebra when p(m) = q(c+2) for m is an element of Z, they commute w hen in addition p = q(2k) for k integer non-zero, and they belong to t he center of A(q,p)(<(sl)over cap>(2)(c)) when k is odd. The Poisson s tructures obtained for t(z) in these classical limits contain the q-de formed Virasoro algebra, characterizing the structures at p not equal q2k as new W-q,W-p(sl(2)) algebras.