Types of elements in non-commutative Poisson algebras and Dixmier Conjecture

Non-commutative Poisson algebras are the algebras having an associative algebra structure and a Lie algebra structure together with the Leibniz law. Let $P$ be a non-commutative Poisson algebra over some algebraically closed field of characteristic zero. For any $z\in P$, there exist four subalgebras of $P$ associated with the inner derivation $ad_z$ on $P$. Based on the relationships between these four subalgebras, elements of $P$ can be divided into eight types. We will mainly focus on two types of non-commutative Poisson algebras: the usual Poisson algebras and the associative algebras with the commutator as the Poisson bracket. The following problems are studied for such non-commutative Poisson algebras: how the type of an element changes under homomorphisms between non-commutative Poisson algebras, how the type of an element changes after localization, and what the type of the elements of the form $z_1 \otimes z_2$ and $z_1 \otimes 1 + 1 \otimes z_2$ is in the tensor product of non-commutative Poisson algebras $P_1\otimes P_2$. As an application of above results, one knows that Dixmier Conjecture for $A_1$ holds under certain conditions. Some properties of the Weyl algebras are also obtained, such as the commutativity of certain subalgebras.