Abstract
It is shown that the elliptic algebra \(\mathcal{A}_{q,p} (\widehat{sl}(2)_c )\) at the critical level c = −2 has a multidimensional center containing some trace-like operators t(z). A family of Poisson structures indexed by a non-negative integer and containing the q-deformed Virasoro algebra is constructed on this center. We show also that t(z) close an exchange algebra when p m = q c+2 for \(m \in \mathbb{Z}\), they commute when in addition p = q 2k for k integer non-zero, and they belong to the center of \(\mathcal{A}_{q,p} (\widehat{sl}(2)_c )\) when k is odd. The Poisson structures obtained for t(z) in these classical limits contain the q-deformed Virasoro algebra, characterizing the structures at p ≠ q 2k as new \(\mathcal{W}_{q,p} (sl(2))\) algebras.
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Avan, J., Frappat, L., Rossi, M. et al. From quantum to elliptic algebras. Czechoslovak Journal of Physics 47, 1083–1092 (1997). https://doi.org/10.1023/A:1021645814342
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DOI: https://doi.org/10.1023/A:1021645814342