R-matrix Quantization of the Ruijsenaars-Schneider Models

I describe an algebraic scheme for quantizing the Ruijsenaars-Schneider
models in the R-matrix formalism. It is based on a special parametrization
of the cotangent bundle over GL(n,C). In new variables
the standard symplectic structure is described by a classical (Frobenius)
r-matrix and by a new dynamical r¯-matrix. Quantizing these
r-matrices, I will exhibit the quantum L-operator algebra and construct
its particular representation corresponding to the RuijsenaarsSchneider
system. I will also indicate a couple of open problems.