Structure constants in Poisson algebras

Question

I am currently studying Poisson algebras. Regarding the structure constants of a Poisson algebra, How can it be defined for Poisson algebras?

Answer 1

If your Poisson algebra $A$ is generated as an algebra by $x^1,\ldots,x^n$, then the Poisson bracket of arbitrary elements of $A$ can be expressed (using bi-linearity and the bi-derivation property) in terms of the structure functions $P^{ij}=\{x^i, x^j\} \in A$.

Of course these are antisymmetric $P^{ji} =-P^{ij}$ and the Jacobi identity for $(f,g,h) =(x^i,x^j,x^k)$ leads to $$\sum_{\ell=1}^n P^{i\ell}\partial_\ell P^{jk} + P^{j\ell}\partial_\ell P^{ki}+P^{k\ell}\partial_\ell P^{ij}=0 \qquad \text{ for all }\quad 1 \leqslant i <j<k\leqslant n.$$

Here I used the abbreviation $\partial_\ell = \frac{\partial}{\partial x^\ell}$.

Conversely, a set of structure functions $P^{ij}$ satisfying these equations can be used to define a Poisson bracket, $$\{f,g\}=\sum_{i,j=1}^n P^{ij}\cdot \partial_i f \cdot \partial_j g$$

If you put the structure functions into a matrix, it is called the Poisson matrix.

In the linear case $P^{ij} = \sum_k c^{ij}_k x^k$ the structure functions are completely determined by the constants $c^{ij}_k$, but a general Poisson algebra cannot be captured using only a finite set of constants.

For a reference, see e.g. the book Poisson Structures by Laurent-Gengoux, Pichereau, and Vanhaecke, in particular section 1.2.2.