The Starred Dixmier Conjecture for A₁

Abstract

Let A ₁(K) = K ⟨ X, Y | YX − XY = 1 ⟩ be the first Weyl algebra over a characteristic zero field K, and let α be the exchange involution on A ₁(K) given by α(X) = Y and α(Y) = X. The Dixmier conjecture of Dixmier (1968) asks the following question: Is every algebra endomorphism of the Weyl algebra A ₁(K) an automorphism? The aim of this paper is to prove that each α-endomorphism of A ₁(K) is an automorphism. Here an α-endomorphism of A ₁(K) is an endomorphism which preserves the involution α. We also prove an analogue result for the Jacobian conjecture in dimension 2, called α −JC ₂.

Key Words:

• Dixmier conjecture
• Involution
• Jacobian conjecture

2010 Mathematics Subject Classification:

• Primary: 16W20
• 14R15
• Secondary: 16S32

Notes

Communicated by I. Shestakov.