Remark on the Integrals of Motion Associated with Level $k$ Realization of the Elliptic Algebra $U_{q,p} (\widehat{\mathfrak{sl}}_{2})$

Kojima, Takeo; Shiraishi, Jun'ichi

doi:10.7546/giq-10-2009-183-196

VOL. 10 | 2009 Remark on the Integrals of Motion Associated with Level

k

Realization of the Elliptic Algebra

U_{q, p} ({\hat{sl}}_{2})

Chapter Author(s) Takeo Kojima, Jun'ichi Shiraishi

Editor(s) Ivaïlo M. Mladenov, Gaetano Vilasi, Akira Yoshioka

Geom. Integrability & Quantization, 2009: 183-196 (2009) DOI: 10.7546/giq-10-2009-183-196

Abstract

We give one parameter deformation of level $k$ free field realization of the screening current of the elliptic algebra $U_{q, p} ({\hat{sl}}_{2})$ . By means of these free field realizations, we construct infinitely many commutative operators, which are called the nonlocal integrals of motion associated with the elliptic algebra $U_{q, p} ({\hat{sl}}_{2})$ for level $k$ . They are given as integrals involving a product of the screening current and elliptic theta functions. This paper give level $k$ generalization of the nonlocal integrals of motion given in [1].