Normal forms of elements in the Weyl algebra and Dixmier Conjecture

TL;DR

This paper characterizes the normal forms of nilpotent and semisimple elements in the Weyl algebra, introduces the Joseph norm form, and explores implications for the Dixmier and Jacobian conjectures.

Contribution

It establishes a unique normal form for elements in the Weyl algebra and links this to reformulations of the Dixmier and Jacobian conjectures.

Findings

Normal form corresponds to a unique pair of integers (k,n).

The Dixmier conjecture holds if k or n is prime.

Analogous results are obtained for the Jacobian conjecture.

Abstract

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.