Non-local integrals of motion for deformed $W$-algebra $W_{q,t}(g)$ associated with $g=A_l^{(1)}, D_l^{(1)}, E_{6,7,8}^{(1)}$

Abstract

We present an infinite set of non-local integrals of motion for the deformed $W$-algebra $W_{q,t}(g)$ associated with the affine Lie algebras $g=A_l^{(1)}, D_l^{(1)}$ and $E_{6,7,8}^{(1)}$.They can be regarded as a two-parameter deformation of trace of the monodromy matrix of the $g$-KdV theory. Commutativity of the non-local integrals of motion is shown in the case of $g=A_l^{(1)}$ and $D_l^{(1)}$ by a direct calculation. In the case of $g=E_{6,7,8}^{(1)}$ it is a conjecture.