2024/07/16 by Gang Han, Han, Gang, Zhennan Pan +3
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #FOS: Mathematics #Finite Group Theory Research #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2407.11291
openalex publication_date 2024/07/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A result of A. Joseph says that any nilpotent or semisimple element z in the Weyl algebra A1 over some algebracally closed field K of characterstic 0 has a normal form up to the action of the automorphism group of A1. It is shown in this note that the normal form corresponds to some unique pair of integers (k,n) with k≥ n≥ 0, and will be called the Joseph norm form of z. Similar results for the symplectic Poisson algebra S1 are obtained. The Dixmier conjecture can be reformulated as follows: For any nilpotent element z∈ A1 whose Joseph norm corresponds to (k,n) with k>n≥ 1, there exists no w∈ A1 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.