The paper deals with the solutions r(x) of the classical Yang-Baxter equation \[ (1)\quad [r^{12}(x),r^{13}(xy)]+[r^{12}(x),r^{23}(y)]+[r^{13}(xy),r^{23 }(y)]=0, \] for the generalized Toda system of type \(\hat{\mathfrak g}=A_ n^{(1)};B_ n^{(1)};C_ n^{(1)};D_ n^{(1)};A_{2n}^{(2)};A^{(2)}_{2n-1}\) and \(D^{(2)}_{n+1}\), where r(x) is a \({\mathfrak g}\otimes {\mathfrak g}\)-valued rational function, \({\mathfrak g}\) being a finite dimensional simple Lie algebra, and \(r^{12}(x)=r(x)\otimes I\), etc.
The problem is to find an \(R(x)=R(x,h)\) containing an arbitrary parameter h, such that: (i) It satisfies the quantum Yang-Baxter equation \[ (2)\quad R^{12}(x)\quad R^{13}(xy)\quad R^{23}(x)=R^{23}(x)\quad R^{13}(xy)\quad R^{12}(x), \] and (ii) As \(h\to 0\), (3) \(R(x,h)=\kappa (x,h)(I+hr(x)+...)\) holds with scalar \(\kappa(x,h)\).
The main result of the paper is the explicit construction of matrices R(x)\(\in End(V\otimes V)\) satisfying (2), where V is a finite dimensional representation of \({\mathfrak g}\) and \(R(x)\) takes values in \(U({\mathfrak g})\otimes U({\mathfrak g})\), where U(\({\mathfrak g})\) denotes the universal enveloping algebra of \({\mathfrak g}.\)
The paper is very clearly written.