We extend to the sl(N) case the results that we previously obtained on the construction of W_{ q,p} algebras from the elliptic algebra A_{ q,p}(\hat{ sl}(2)_c). The elliptic algebra A_{ q,p}(\hat{ sl}(N)_c) at the critical level c=-N has an extended center containing trace-like operators t(z). Families of Poisson structures indexed by N(N-1)/2 integers, defining q-deformations of the W_N algebra, are constructed. The operators t(z) also close an exchange algebra when (-p^1/2)^{ NM} = q^{ -c-N} for M in Z. 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 algebras depending on the parity of h, characterizing the exchange structures at p \ne q^{ Nh} as new W_{ q,p}(sl(N)) algebras.
Categories: math.QA hep-th nlin.SI solv-int