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Let $A$ be a polynomial algebra over a field of characteristic $0$ in the variables $x_1,\dots,x_n$. Consider polynomials $f_1,\dots,f_m\in A$ and let $I$ be the ideal they generate in $A$. Moreover, assume that $\{\:,\:\}$ is a Poisson bracket on $A$, i.e., that it is a Lie bracket on $A$ and a derivation in each argument. We say that $I$ is a Poisson ideal in A if $\{I,A\}\subset I$. This is equivalent to saying that there exist polynomials $Z_{ij}^k\in A$ such that \begin{align} \{x_i,f_j\}=\sum_{k=1}^m Z_{ij}^k f_k, \end{align} where $j\in\{1,\dots m\}$ and $i\in\{1,\dots n\}$. In this case the bracket descends to $A/I$ and we say that $A/I$ is an affine Poisson algebra.

So far my attempts to construct examples with nonzero $Z_{ij}^k$ have failed. My question is: are there examples with nonzero $Z_{ij}^k$? In the case when the Poisson structure is constant or linear there might be conceptual reasons for the vanishing of the $Z_{ij}^k$. Does anybody know of results in this direction?

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    $\begingroup$ I correct myself: I am seeking for examples with $Z_{ij}^k$ not in $I$. In other words: the class of $Z_{ij}^k$ in $A/I$ should be nonzero. $\endgroup$ – HCH Mar 6 at 21:23
  • $\begingroup$ Lie ideals yield Poisson ideals of the Lie-Poisson structure; look for example in the upper triangular matrices. $\endgroup$ – Ricardo Buring Mar 16 at 13:57
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Let $L$ be a finite-dimensional non-abelian Lie algebra over a field $\mathbb{F}$ and consider the symmetric algebra $S(L)$ of $L$, which you can identify with the polynomial ring $\mathbb{F}[x_1,x_2,\ldots]$ where $x_1,x_2,\ldots,x_n$ is an $\mathbb{F}$-basis of $L$ over $\mathbb{F}$. Then the Lie bracket of $L$ can be uniquely extended to a Poisson bracket of $S(L)$ so that this commutative algebra becomes a Poisson algebra. Now, take an ideal $I$ of $L$ and note that $J=I\cdot S(L)$ is a Poisson ideal of $S(L)$. By using the structure constants of $L$, you can now easily find a lot of examples of the kind you are looking for.

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Lie ideals in Lie algebras also define Poisson ideals of the associated Lie-Poisson structure.

Consider the Lie-Poisson structure associated to the Lie algebra of upper-triangular $3\times 3$ matrices. The Poisson structure matrix with respect to the generators $E_{11}, E_{12}, E_{13}, E_{22}, E_{23}, E_{33}$ is given by$$\left(\begin{array}{rrrrrr} 0 & E_{12} & E_{13} & 0 & 0 & 0 \\ -E_{12} & 0 & 0 & E_{12} & E_{13} & 0 \\ -E_{13} & 0 & 0 & 0 & 0 & E_{13} \\ 0 & -E_{12} & 0 & 0 & E_{23} & 0 \\ 0 & -E_{13} & 0 & -E_{23} & 0 & E_{23} \\ 0 & 0 & -E_{13} & 0 & -E_{23} & 0 \end{array}\right)$$ We have for example $\{E_{13},E_{11}\} = -E_{13}$ and $\{E_{13},E_{33}\} = E_{13}$ and the other brackets with $E_{13}$ are zero, so $I = \langle E_{13} \rangle$ is a Poisson ideal, and the coefficients in the nontrivial relations are $\pm 1 \not\in I$.

In a Poisson algebra with a Lie-Poisson structure we can also form the ideal generated by all monomials of degree $2$. Continuing example above we have e.g. $$\{E_{11}E_{33},E_{13}\}=E_{11}\{E_{33},E_{13}\}+E_{33}\{E_{11},E_{13}\} = -E_{11}E_{13}+E_{33}E_{13},$$ so again there are nontrivial relations with constant coefficients which do not belong to the ideal.

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