Schlesinger Transformations and Quantum R-Matrices

 Schlesinger transformations are discrete monodromy preserving symmetry transformations of a meromorphic connection which shift by integers the eigenvalues of its residues. We study Schlesinger transformations for twisted \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document}-valued connections on the torus. A universal construction is presented which gives the elementary two-point transformations in terms of Belavin's elliptic quantum R-matrix. In particular, the role of the quantum deformation parameter is taken by the difference of the two poles whose residue eigenvalues are shifted. Elementary one-point transformations (acting on the residue eigenvalues at a single pole) are constructed in terms of the classical elliptic r-matrix.