From quantum to elliptic algebras
Résumé
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.