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A Remark on the Dixmier Conjecture
- Part of
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- Journal:
- Canadian Mathematical Bulletin / Volume 63 / Issue 1 / March 2020
- Published online by Cambridge University Press:
- 30 August 2019, pp. 6-12
- Print publication:
- March 2020
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- Article
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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 
$Q$).
