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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2407.11291 |
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Table of Contents:
- A result of A. Joseph says that any nilpotent or semisimple element $z$ in the Weyl algebra $A_1$ over some algebracally closed field $K$ of characterstic 0 has a normal form up to the action of the automorphism group of $A_1$. It is shown in this note that the normal form corresponds to some unique pair of integers $(k,n)$ with $k\ge n\ge 0$, and will be called the Joseph norm form of $z$. Similar results for the symplectic Poisson algebra $S_1$ are obtained. The Dixmier conjecture can be reformulated as follows: For any nilpotent element $z\in A_1$ whose Joseph norm corresponds to $(k,n)$ with $k>n\ge 1$, there exists no $w\in A_1$ 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.