The Starred Dixmier Conjecture forA₁

Abstract

Let A 1 K = K X Y YX − XY = 1 be the first Weyl algebra over a characteristic zero field K, and let be the exchange involution on A 1 K given by X = Y and 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 -endomorphism of A 1 K is an automorphism. Here an -endomorphism of A 1 K is an endomorphism which preserves the involution . We also prove an analogue result for the Jacobian conjecture in dimension 2, called − JC 2 .

About Dixmier's conjecture

“…This result can be obtained in a similar way to the way the result concerning α-endomorphisms was obtained in [9].…”

“…I wish to thank Prof. C. Valqui for working with me on the starred Dixmier conjecture [9], Prof. J. Bell for his note [3] concerning involutions on A 1 , and Prof. L. Rowen and Prof. U. Vishne for listening to my ideas and commenting on them.…”

“…Let K[x, y] be the polynomial ring over K and let α be the exchange involution on K[x, y] given by α(x) = y and α(y) = x. The α-Jacobian conjecture (α − JC 2 ) was posed and proved: If f : K[x, y] → K[x, y] is an α-morphism (f • α = α • f ) that satisfies Jac(f (x), f (y)) = 1, then f is invertible, [9,Proposition 4.1].…”

“…In [9] only two involutions were considered: α and β, where β is given by β(X) = X and β(Y ) = −Y . Clearly, β is conjugate to α by ϕ: β := ϕ Definition 2.2 (A γ, δ-endomorphism).…”

Observations on the two dimensional Jacobian Conjecture

“…Our proofs of Theorem 3.3 and Theorem 3.4 rely on results found in [18] and [19]. For the convenience of the reader, we now bring the specific results which are needed in the proofs of Theorem 3.3 and Theorem 3.4.…”