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.