TITLE: Is there a classification of polynomial Poisson brackets?
QUESTION [11 upvotes]: As an example, consider the following Poisson bracket on ${\mathbb R}^n$:
$$\{x_i, x_{i+1}\} = x_ix_{i+1}(x_i+x_{i+1}),\\
\{x_i, x_{i+2}\} = x_ix_{i+1}x_{i+2}.$$
The indices are taken modulo $n$, and the "distant" variables commute. The Jacobi identity holds but does not look obvious. This bracket appears in the study of Volterra lattice or discrete KdV.
Is there a classification of Poisson brackets whose values on coordinate functions are given by polynomials in the coordinates? Or are some large families of such brackets known? What if we require that the cyclic permutation of coordinates is a Poisson map, as in the above example?

REPLY [2 votes]: This is a late partial answer, but I agree with the previous answer in that I don't think there are many general classifications of polynomial Poisson structures available. The only classification for higher-order Poisson structures that I have been able to find is the case of (homogeneous) quadratic Poisson structures in dimension 3 due to Dufour and Haraki (MR1086519, available here), which are classified into 14 families of Poisson structures. This suggests that a classification in higher degrees or in higher dimensions will be quite involved.