Algebraic Properties of Compatible Poisson Brackets

摘要

We discuss algebraic properties of a pencil generated by two compatible Poisson tensors A(x) and B(x). From the algebraic viewpoint this amounts to studying the properties of a pair of skew-symmetric bilinear forms A and B defined on a finite-dimensional vector space. We describe the Lie group G_P of linear automorphisms of the pencil P = {A + λB}. In particular, we obtain an explicit formula for the dimension of G_P and discuss some other algebraic properties such as solvability and Levi –Malcev decomposition.