Some progress in the Dixmier Conjecture

TL;DR

This paper advances the understanding of the Dixmier Conjecture by identifying conditions under which elements generate the Weyl algebra, utilizing spectral analysis and Newton polygons to establish new criteria.

Contribution

It provides new sufficient conditions involving spectral components for elements to generate the Weyl algebra, and generalizes previous results using Newton polygons.

Findings

If either element lacks negative spectrum components, the conjecture holds.

Introduces criteria based on spectral properties for generation of the algebra.

Utilizes Newton polygons as a key analytical tool.

Abstract

Let $p$ and $q$, where $pq-qp=1$, be the standard generators of the first Weyl algebra $A_1$ over a field of characteristic zero. Then the spectrum of the inner derivation $ad(pq)$ on $A_1$ are exactly the set of integers. The algebra $A_1$ is a $\mathbb{Z}$-graded algebra with each $i$-component being the $i$-eigenspace of $ad(pq)$, where $i\in \mathbb{Z}$. The Dixmier Conjecture says that if some elements $z$ and $w$ of $A_1$ satisfy $zw-wz=1$, then they generate $A_1$. We show that if either $z$ or $w$ possesses no component belonging to the negative spectrum of $ad(pq)$, then the Dixmier Conjecture holds. We give some generalization of this result, and some other useful criterions for $z$ and $w$ to generate $A_1$. An important tool in our proof is the Newton polygon for elements in $A_1$.