Isomonodromic deformations, generalized Knizhnik–Zamolodchikov equations and non-skew-symmetric classical r-matrices

Abstract

Using non-skew-symmetric classical r-matrices r12(u,v) with spectral parameters, we generalize Fuchsian systems and Schlesinger equations (isomonodromy equations) viewing the last as the nonautonomous Hamiltonian equations corresponding to the generalized Gaudin Hamiltonians. Quantizing these systems we construct generalization of Knizhnik–Zamolodchikov equations corresponding to the non-skew-symmetric classical r-matrices r12(u,v) with spectral parameters. In the case of an ordinary skew-symmetric classical r-matrix r12(u−v) depending on the difference of the spectral parameters, we reobtain, as special partial cases, a standard generalization of Fuchsian systems, Schlesinger, and Knizhnik–Zamolodchikov equations.