About Dixmier's conjecture

2014/06/17 by Moskowicz, Vered
#FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1406.4368

Abstract

The well-known Dixmier conjecture 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. Up to checking that in the group generated by automorphisms and anti-automorphisms of A₁ all the involutions belong to one conjugacy class, we show that every involutive endomorphism from (A₁,γ) to (A₁,δ) is an automorphism (γ and δ are two involutions on A₁), and 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.

Related

• The starred Dixmier's conjecture
• The starred Dixmier conjecture for A₁
• The Conjecture of Dixmier for the first Weyl algebra is true
• The Dixmier conjecture and the shape of possible counterexamples II
• An analogue of the Conjecture of Dixmier is true for the algebra of polynomial integro-differential operators