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

Research paper by **V. V. Bavula**

Indexed on: **02 May '11**Published on: **02 May '11**Published in: **Mathematics - Rings and Algebras**

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#### 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?}).