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Remark on the integrals of motion associated with level \(k\) realization of the elliptic algebra \(U_{q,p}(\widehat{\mathfrak{sl}_2})\). (English) Zbl 1230.17009

Summary: We give one parameter deformation of level \(k\) free field realization of the screening current of the elliptic algebra \(U_{q,p}(\widehat{\mathfrak{sl}_2})\). 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 \(U_{q,p}(\widehat{\mathfrak{sl}_2})\) for level \(k\). They are given as integrals involving a product of the screening current and elliptic theta functions. This paper gives a level \(k\) generalization of the nonlocal integrals of motion given in [B. Feigin, T. Kojima, J. Shiraishi and H. Watanabe, The integrals of motion for the deformed Virasoro algebra. http://arxiv.org/abs/0705.0427].

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
33D80 Connections of basic hypergeometric functions with quantum groups, Chevalley groups, \(p\)-adic groups, Hecke algebras, and related topics