About Dixmier's conjecture

Abstract

The well-known Dixmier conjecture [5] asks if every algebra endomorphism of the first Weyl algebra over a characteristic zero field is an automorphism. We bring a hopefully easier to solve conjecture, called the γ, δ conjecture, and show that it is equivalent to the Dixmier conjecture. In the group generated by automorphisms and anti-automorphisms of A1, all the involutions belong to one conjugacy class, hence: • Every involutive endomorphism from (A1, γ) to (A1, δ) is an automorphism (γ and δ are two involutions on A1). • Given an endomorphism f of A1 (not necessarily an involutive endomorphism), if one of f(X), f(Y) is symmetric or skew-symmetric (with respect to any involution on A1), then f is an automorphism.