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Deformed W_N algebras from elliptic sl(N) algebras

J. Avan·L. Frappat·M. Rossi·P. Sorba
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Abstract

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.

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J. Avan, L. Frappat, M. Rossi, P. Sorba. 1998-01-22. Deformed W_N algebras from elliptic sl(N) algebras. https://doi.org/10.1007/s002200050517

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