A Remark on the Dixmier Conjecture. Issue 1 (30th March 2020)

• Record Type: Journal Article
• Authors: Bavula, V. V.; Levandovskyy, V.
• Is Part Of: Canadian mathematical bulletin =. Volume 63:Issue 1(2020)
• Page Start: 6
• Page End: 12
• Publication Date: 2020-03-30
• Subjects: 16S50, -- 16W20, -- 16S32, -- 16W50; Weyl algebra, -- Dixmier Conjecture, -- automorphism, -- endomorphism, -- a ℤ-graded algebra; Mathematics -- Periodicals; Mathematics; Periodicals; 510.5
• DOI: 10.4153/S0008439519000122
• Languages: English
• ISSNs: 0008-4395

Abstract

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 $Q$ ).