On quadratic Poisson structures (Q1202950)

Quadratic Poisson structures on a finite-dimensional vector space are considered, i.e. those Poisson structures where the Poisson brackets of coordinate functions are homogeneous quadratic polynomials. The main result of this paper is that each quadratic Poisson structure arises from a special construction out of a Poisson \(G\)-space and a classical triangular \(r\)-matrix. Such a construction is called in this paper the canonical decomposition of a Poisson structure. For the cases of low- dimensional underlying vector space (two- or three-dimensional) the canonical decomposition is used to obtain the full classification of quadratic Poisson structures up to a Poisson diffeomorphism.