2022/10/01 by Gang Han, Bowen Tan, Han, Gang +1
Mathematics · Computer Science · #Advanced Differential Equations and Dynamical Systems #Advanced Topics in Algebra #Coding theory and cryptography
paper · pdf · doi:10.48550/arxiv.2210.00257
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 ℤ-graded algebra with each i-component being the i-eigenspace of ad(pq), where i∈ ℤ. The Dixmier Conjecture says that if some elements z and w of A1 satisfy zw-wz=1, then they generate A1. 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 A1. An important tool in our proof is the Newton polygon for elements in A1.