arXiv subjects

Zhennan Pan

Publications and source records attributed to Zhennan Pan.

3 recordsLinked to original sources

arXiv2025

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.

math.RA↗

arXiv2024

Normal forms of elements in the Weyl algebra and Dixmier Conjecture

A result of A. Joseph says that any nilpotent or semisimple element $z$ in the Weyl algebra $A_1$ over some algebracally closed field $K$ of characterstic 0 has a normal form up to the action of the automorphism group of $A_1$. It is shown in this note that the normal form corresponds to some unique pair of integers $(k,n)$ with $k\ge n\ge 0$, and will be called the Joseph norm form of $z$. Similar results for the symplectic Poisson algebra $S_1$ are obtained. The Dixmier conjecture can be reformulated as follows: For any nilpotent element $z\in A_1$ whose Joseph norm corresponds to $(k,n)$ with $k>n\ge 1$, there exists no $w\in A_1$ with $ [z,w]=1$. It is known to hold true if $k$ and $n$ are coprime. In this note we show that the assertion also holds if $k$ or $n$ is prime. Analogous results for the Jacobian conjecture for $K[X,Y]$ are obtained.

math.RA↗

arXiv2024

Deformed Laurent series rings and completions of the Weyl division ring

Let $ L((T^{-1}))$ be the space of (inverse) Laurent serieswith coefficients in some field $L$. It has a standard degree map and the induced topology. With its usual addition and a new product on this space which is continuous and preserves the standard degree map, it will be a complete topological division ring, and called a deformed Laurent series ring. Under mild restrictions, we give the necessary and sufficient conditions for a product on $ L((T^{-1}))$ to make it a deformed Laurent series ring. Then we apply the above theory to construct the completions of the Weyl division ring $D_1$, over some field of characteristic 0, with respect to a class of discrete valuations on it. Such completions are topological division rings with nice properties. For instance, their valuation rings are non-commutative Henselian rings; the centralizer of each element not in the center is commutative.

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