From Quantum to Elliptic Algebras on Leave of Absence from Laboratoire De Physique Théorique Enslapp

It is shown that the elliptic algebra Aq,p(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 ∈ Z, they commute when in addition p = q 2k for k integer non-zero, and they belong to the center of Aq,p(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 Wq,p(sl(2)) algebras.