From quantum to elliptic algebras

Details From quantum to elliptic algebras From quantum to elliptic algebras

1997-07-29

arxiv.org/abs/q-alg/9707034v2

Mathematics

Quantum Algebra

LaTeX2e Document - packages subeqn,amsfonts

Scientific paper

10.1023/A:1021645814342

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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