Poisson homology of r-matrix type orbits I:: Example of computation

In this paper we consider the Poisson algebraic structure associated with a classical r-matrix, i.e. with a solution of the modified classical Yang-Baxter equation. In Section 1 we recall the concept and basic facts of the r-matrix type Poisson orbits. Then we describe the r-matrix Poisson pencil (i.e the pair of compatible Poisson structures) of rank 1 or CPn-type orbits of SL(n, C). Here we calculate symplectic leaves and the integrable foliation associated with the pencil. We also describe the algebra of functions n CPn-type orbits. In Section 2 we calculate the Poisson homology of Drinfeld-Sklyanin Poisson brackets which belong to the r-matrix Poisson family.