q-deformed W-algebras and elliptic algebras

The elliptic algebra Aq,p(ŝl(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 WN algebra, are constructed. The operators t(z) also close an exchange algebra when (−p1/2)NM = q−c−N for M ∈ Z. It becomes Abelian when in addition p = qNh where h is a non-zero integer. The Poisson structures obtained in these classical limits contain different q-deformed WN algebras depending on the parity of h, characterizing the exchange structures at p 6= qNh as new Wq,p(sl(N)) algebras. MSC number: 81R50, 17B37 Talk presented by L. Frappat at the 7th International Colloquium “Quantum Groups and Integrable Systems” (QGIS), Prague, 18-20 June 1998. math.QA/9807131 July 1998