Poisson closures and Poisson $A$-algebras

We study the interaction between the Lie and Poisson subalgebras of a Poisson algebra. Given a Lie subalgebra $U$ of the underlying Lie algebra $\mathcal P_L$, we consider the Poisson subalgebra generated by $U$, called its {Poisson closure}, and investigate when Lie-theoretic properties such as nilpotency and solvability are preserved under this construction. We also study the stability under Poisson closure of several distinguished Lie subalgebras, including maximal nilpotent and solvable subalgebras, Engel and Cartan subalgebras and the derived algebra. As an application, we investigate Poisson $A$-algebras, defined by the condition that every nilpotent subalgebra is a zero algebra. In the finite-dimensional case, we obtain structural results relating Poisson $A$-algebras to Lie $A$-algebras and to the associative structure of the algebra.