arXiv math/98121461998-12math.DGmath.QAnlin.SI

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

Alexei Kotov

发表信息：: J. Nonlinear Math. Phys. 6 (1999), no. 4, 365-383
DOI：: 10.2991/jnmp.1999.6.4.2

英文摘要：

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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