# Nontrivial Poisson relations for affine Poisson algebras

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?

• 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.
– HCH
Mar 6, 2020 at 21:23
• Lie ideals yield Poisson ideals of the Lie-Poisson structure; look for example in the upper triangular matrices. Mar 16, 2020 at 13:57

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.
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.