RMatrix Quantization of the Elliptic RuijsenaarsSchneider Model
Abstract
It is shown that the classical Loperator algebra of the elliptic RuijsenaarsSchneider model can be realized as a subalgebra of the algebra of functions on the cotangent bundle over the centrally extended current group in two dimensions. It is governed by two dynamical r and ?rmatrices satisfying a closed system of equations. The corresponding quantum R and ?Rmatrices are found as solutions to quantum analogs of these equations. We present the quantum Loperator algebra and show that the system of equations on R and ?R arises as the compatibility condition for this algebra. It turns out that the Rmatrix is twistequivalent to the Felder elliptic R^{F}matrix with ?R playing the role of the twist. The simplest representation of the quantum Loperator algebra corresponding to the elliptic RuijsenaarsSchneider model is obtained. The connection of the quantum Loperator algebra to the fundamental relation RLL=LLR with Belavin's elliptic R matrix is established. As a byproduct of our construction, we find a new Nparameter elliptic solution to the classical YangBaxter equation.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 1998
 DOI:
 10.1007/s002200050303
 Bibcode:
 1998CMaPh.192..405A