The Poisson center and polynomial, maximal Poisson commutative subalgebras, especially for nilpotent Lie algebras of dimension at most seven

Let g be a finite dimensional Lie algebra over an algebraically closed field k of characteristic zero. We collect some general results on the Poisson center of S(g), including some simple criteria regarding its polynomiality, and also on certain Poisson commutative subalgebras of S(g). These facts are then used to complete our previous work on the subject (Ooms, 2009 [O5, 5]), i.e. to give an explicit description for the Poisson center of all indecomposable, nilpotent Lie algebras of dimension at most seven. Among other things, we also provide a polynomial, maximal Poisson commutative subalgebra of S(g), enjoying additional properties. As a by-product we show that a conjecture by Milovanov is valid in this situation. These results easily carry over to the enveloping algebra U(g).