On the centralizers in the Weyl algebra

TL;DR

This paper proves that in the Weyl algebra, the centralizer of an element P with a specific commutation relation is exactly the polynomial algebra generated by P.

Contribution

It establishes that if two elements in the Weyl algebra satisfy a particular commutation relation, then the centralizer of one is a polynomial algebra in that element, clarifying the structure of centralizers.

Findings

Centralizer of P is the polynomial algebra k[P] when [Q,P]=1

Provides a characterization of centralizers in the Weyl algebra

Advances understanding of algebraic structures in Weyl algebra

Abstract

Let P,Q be elements of the Weyl algebra W. We prove that if [Q,P]=1, then the centralizer of P is the polynomial algebra k[P].