On the eigenvector algebra of the product of elements with commutator one in the first Weyl algebra

TL;DR

This paper investigates the eigenvalues and eigenvector algebra of the inner derivation in the first Weyl algebra, confirming the eigenvalues are integers and characterizing the eigenvector algebra under algebra homomorphisms.

Contribution

It proves that the eigenvalues of the inner derivation in the Weyl algebra are integers and identifies the eigenvector algebra as generated by the elements x and y.

Findings

Eigenvalues of the inner derivation are integers (Z).

Eigenvector algebra is generated by x and y.

Results relate to the open Dixmier conjecture.

Abstract

Let $A_1=K < X, Y | [Y,X]=1>$ be the (first) Weyl algebra over a field $K$ of characteristic zero. It is known that the set of eigenvalues of the inner derivation $\ad (YX)$ of $A_1$ is $\Z$. Let $ A_1\ra A_1$, $X\mapsto x$, $Y\mapsto y$, be a $K$-algebra homomorphism, i.e. $[y,x]=1$. It is proved that the set of eigenvalues of the inner derivation $\ad (yx)$ of the Weyl algebra $A_1$ is $\Z$ and the eigenvector algebra of $\ad (yx)$ is $K< x,y> $ (this would be an easy corollary of the Problem/Conjecture of Dixmier of 1968 [still open]: {\em is an algebra endomorphism of $A_1$ an automorphism?}).