Nontrivial Poisson relations for affine Poisson algebras

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

I apologize for the slow processing. The case of Poisson ideals coming from linear Poisson structures have been systematically studied in the literature. Similarly there are the symplectic reductions. For details and examples one may consult the recent https://arxiv.org/pdf/2107.04204.pdf For polynomial Poisson structures of higher degree I cannot find any systematic study in the literature.