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A Remark on the Dixmier Conjecture

Part of: Rings and algebras with additional structure Rings and algebras arising under various constructions

Published online by Cambridge University Press:  30 August 2019

V. V. Bavula  and
V. Levandovskyy
V. V. Bavula
Affiliation:
Department of Pure Mathematics, University of Sheffield, Hicks Building, SheffieldS3 7RH, UK Email: v.bavula@sheffield.ac.uk
V. Levandovskyy
Affiliation:
Lehrstuhl D für Mathematik, RWTH Aachen University, 52062 Aachen, Germany Email: Viktor.Levandovskyy@math.rwth-aachen.de
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Abstract

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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 $Q$).

Keywords

Weyl algebraDixmier Conjectureautomorphismendomorphisma ℤ-graded algebra

MSC classification

Primary: 16S50: Endomorphism rings; matrix rings
Secondary: 16W20: Automorphisms and endomorphisms 16S32: Rings of differential operators 16W50: Graded rings and modules

Information

Type
Article
Information
Canadian Mathematical Bulletin , Volume 63 , Issue 1 , March 2020 , pp. 6 - 12
Copyright
© 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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