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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?

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    $\begingroup$ A large class of examples is given by Poisson algebras of the form $S(\mathfrak{g})$ where $\mathfrak{g}$ is a finite-dimensional Lie algebra and the Poisson bracket is given by extending the Lie bracket. This reflects a Poisson manifold (even variety) structure on the dual $\mathfrak{g}^{\ast}$, the leaves of which are the coadjoint orbits of $\mathfrak{g}$. $\endgroup$ – Qiaochu Yuan Jul 22 '18 at 3:59
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I don't think we have anything like a classification. In the book Poisson Structures (2013) by Laurent-Gengoux, Pichereau, and Vanhaecke, it is written

"For higher order Poisson structures, starting with quadratic Poisson structures, there is no general theory and there is no immediate interpretation, as there is in the case of constant Poisson structures (in terms of bivectors) and in the case of linear Poisson structures (in terms of Lie brackets)."

on page 232 in Chapter 8.5. The special case of quadratic Poisson structures on $\mathbb{C}^3$ are classified in Chapter 9.2.3 of the book.

A reference I particularly like for polynomial Poisson brackets is Integrable Systems in the Realm of Algebraic Geometry (2001) by Vanhaecke. This book includes further examples: both explicit particular examples and infinite families (including the Poisson-Lie structure given by Qiaochu in the comments).

Since I typically only think about Poisson structures in the Gekhtman-Shapiro-Vainshtein approach to cluster algebras, I am particularly fond of "log-canonical" Poisson brackets. That is, the quadratic bracket given by $\{x_i, x_j\} = c_{ij} x_i x_j$ for some skew-symmetric matrix $(c_{ij})$ of scalars. I have also seen this bracket under other names like "diagonal Poisson structure" for example.

In joint work with Nicholas Ovenhouse, we were considering when polynomial brackets could be reduced by a rational change of coordinates to a polynomial bracket of smaller degree. In looking for examples, we considered dimension 3, where all bracket functions are monomials. It's just an exercise in the Jacobi identity to see that $$\{x,y\} = A(x^iy^jz^k)z^{a-k}$$ $$\{x,z\} = B(x^iy^jz^k)y^{b-j}$$ $$\{y,z\} = C(x^iy^jz^k)x^{c-k}$$ gives a Poisson structure. Even here in dimension 3, I don't know what happens when we allow bracket functions to be more than just monomials.

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