Deformed Algebras from Elliptic sl(N) Algebras

We extend to the sl(N)sl(N) case the results that we previously obtained on the construction of W_q,p{\cal W}_{q,p} algebras from the elliptic algebra A_q,p(^(sl)](2)_c){\cal A}_{q,p}(\widehat{sl}(2)_{c}). The elliptic algebra \elp\elp at the critical level c= m N has an extended center containing trace-like operators t(z). Families of Poisson structures indexed by N(Nу)/2 integers, defining q-deformations of the W_N{\cal W}_{N} algebra, are constructed. The operators t(z) also close an exchange algebra when (-p^\sfrac12)^NM = q^-c-N(-p^\sfrac{1}{2})^{NM} = q^{-c-N} for M ? \ZZM\in\ZZ. It becomes Abelian when in addition p= q^Nh, where h is a non-zero integer. The Poisson structures obtained in these classical limits contain different q-deformed W_N{\cal W}_{N} algebras depending on the parity of h, characterizing the exchange structures at p p q^Nh as new W_q,p(sl(N)){\cal W}_{q,p}(sl(N)) algebras.