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A Remark on the Dixmier Conjecture
Published online by Cambridge University Press: 30 August 2019
Abstract
The Dixmier Conjecture says that every endomorphism of the (first) Weyl algebra $A_{1}$ (over a field of characteristic zero) is an automorphism, i.e., if $PQ-QP=1$ for some $P,Q\in A_{1}$, then $A_{1}=K\langle P,Q\rangle$. The Weyl algebra $A_{1}$ is a $\mathbb{Z}$-graded algebra. We prove that the Dixmier Conjecture holds if the elements $P$ and $Q$ are sums of no more than two homogeneous elements of $A_{1}$ (there is no restriction on the total degrees of $P$ and $Q$).
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- © Canadian Mathematical Society 2019
Footnotes
Author V. V. B. was supported by Graduiertenkolleg “Experimentelle und konstruktive Algebra” of the German Research Foundation (DFG). Author V. L. was supported by Project II.6 of SFB-TRR 195 “Symbolic Tools in Mathematics and their Applications” of the DFG.
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