From quantum to elliptic algebras

J. Avan; L. Frappat; M. Rossi; P. Sorba

doi:10.1023/a:1021645814342

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arXiv · q-alg/9707034

From quantum to elliptic algebras

J. Avan·L. Frappat·M. Rossi·P. Sorba

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Abstract

It is shown that the elliptic algebra ${\cal A}_{q,p}(\hat{sl}(2)_c)$ 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 integer, they commute when in addition p=q^{2k} for k integer non-zero, and they belong to the center of ${\cal A}_{q,p}(\hat{sl}(2)_c)$ 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 generic values of p, q and m as new ${\cal W}_{q,p}(sl(2))$ algebras.

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J. Avan, L. Frappat, M. Rossi, P. Sorba. 1997-07-30. From quantum to elliptic algebras. https://doi.org/10.1023/a%3A1021645814342

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