Algebraic Properties of Compatible Poisson Brackets

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