| Résume | Let L be a finite dimensional Lie algebra over a field k of characteristic zero and let S(L) be its symmetric algebra, equipped with its natural Poisson structure. We collect some general facts on the Poisson center of S(L), including some simple criteria regarding its polynomiality, and also on certain Poisson commutative subalgebras of S(L). These facts allow us to give an explicit description for the Poisson center for all complex, nilpotent Lie algebras of dimension at most seven. Among other things, we also provide in each case a polynomial, maximal Poisson commutative subalgebra of S(L), enjoying additional properties. As a by-product we show that a conjecture by Milovanov is valid in this situation. Finally, all these results easily carry over to the enveloping algebra U(L) of L. |
