q-deformed W-algebras and elliptic algebras

Abstract

The elliptic algebra A_{q,p}(sl(N)_{c}) at the critical level c=-N has an extended center containing trace-like operators t(z). Families of Poisson structures, 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 =/ q^{Nh} as new W_{q,p}(sl(N)) algebras.