The integrals of motion for the elliptic deformation of the Virasoro and image algebra Original Research Article

We review the free field realization of the deformed Virasoro algebra Virq,tVirq,t and the deformed WW albegra View the MathML sourceWq,t(glN̂). We explicitly construct two classes of infinitely many commutative operators ImIm, GmGm, (m∈N)(m∈N), in terms of these algebras. They can be regarded as the elliptic deformation of the local and nonlocal integrals of motion for the conformal field theory. This review is based on the works [B. Feigin, T. Kojima, J. Shiraishi, H. Watanbe, The integrals of motion for the deformed Virasoro algebra (submitted for publication) (2007), Preprint 54 pages, arXiv:0705.0427; B. Feigin, T. Kojima, J. Shiraishi, H. Watanbe, The integrals of motion for the deformed WW algebra View the MathML sourceWq,t(glN̂), in: Proceedings of Representation Theory 2006, Atami, Japan, 2006, pp. 102–114; T. Kojima and J. Shiraishi, The integrals of motion for the deformed WW-algebra View the MathML sourceWq,t(glN̂) II: Proof of the commutation relations, Comm. Math. Phys. 283 (2008) 795–851].