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

Details Poisson homology of r-matrix type orbits I: example of computation Poisson homology of r-matrix type orbits I: example of computation

1998-12-25

arxiv.org/abs/math/9812146v2

J. Nonlinear Math. Phys. 6 (1999), no. 4, 365-383

Mathematics

Differential Geometry

Scientific paper

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 $CP^n$-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 on $CP^n$-type orbits. In Section 2 we calculate the Poisson homology of Drinfeld--Sklyanin Poisson brackets which belong to the $r$-matrix Poisson family.

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