Some progress in the Dixmier conjecture for A 1

Abstract

Let p and q, where p q − q p = 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 Z -graded algebra with each i-component being the i-eigenspace of ad(pq), where i ∈ Z . Assume that z and w are elements of A 1 satisfying z w − w z = 1 . The Dixmier Conjecture for A 1 says that they always generate A 1. We show that if z possesses no component belonging to the negative spectrum of ad(pq), then z and w generate A 1. We give some generalization of this result, and some other useful criterions for z and w to generate A 1. It is shown that if z is a sum of not more than 2 homogeneous elements of A 1 then z and w generate A 1, which generalizes a known result due to Bavula and Levandovskyy.