Some progress in the Dixmier conjecture for A₁

Abstract

Let p and q, where $pq-qp=1$, be the standard generators of the first Weyl algebra A₁ over a field of characteristic zero. Then the spectrum of the inner derivation ad(pq) on A₁ are exactly the set of integers. The algebra A₁ is a $\mathbb{Z}$-graded algebra with each i-component being the i-eigenspace of ad(pq), where $i\in \mathbb{Z}$. Assume that z and w are elements of A₁ satisfying $zw-wz=1$. The Dixmier Conjecture for A₁ says that they always generate A₁. We show that if z possesses no component belonging to the negative spectrum of ad(pq), then z and w generate A₁. We give some generalization of this result, and some other useful criterions for z and w to generate A₁. It is shown that if z is a sum of not more than 2 homogeneous elements of A₁ then z and w generate A₁, which generalizes a known result due to Bavula and Levandovskyy.

KEYWORDS:

• Dixmier conjecture
• Newton polygon
• Weyl algebra

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 16S32
• 16W20

Acknowledgments

We would like to heartily thank Chengbo Wang for his help during the proof of the main result.

Additional information

Funding

Gang Han is supported by Zhejiang Province Science Foundation of China, Grant LY14A010018.