A Remark on the Dixmier Conjecture. Issue 1 (30th March 2020) - Bavula, V. V.; Levandovskyy, V.
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  $Q$ ).