The higher order Hamiltonian structures for the modified classical Yang-Baxter equation

We consider constructing the higher order Hamiltonian structures on the dual of the Lie algebra from the first Hamiltonian structure of the coadjoint orbit method. For this purpose we show that the structure of the Lie algebrag is inherited to the algebra of vector fields ong^* through the solution of the Modified Classical Yang-Baxter equation (Classicalr matrix). We study the algebra that generates the compatible Poisson brackets.This work was supported by Grant Aid for Scientific Research, the Ministry of Education.