About Dixmier's conjecture

TL;DR

This paper introduces the $oldsymbol{ ext{ extgamma, extdelta}}$ conjecture, an equivalent but potentially easier problem to Dixmier's conjecture, and proves automorphism properties for certain involutive endomorphisms of the Weyl algebra.

Contribution

It proposes the $oldsymbol{ ext{ extgamma, extdelta}}$ conjecture, establishes its equivalence to Dixmier's conjecture, and proves automorphism results for involutive endomorphisms with symmetric or skew-symmetric images.

Findings

Every involutive endomorphism between $(A_1, ext{ extgamma})$ and $(A_1, ext{ extdelta})$ is an automorphism.

Endomorphisms with symmetric or skew-symmetric images are automorphisms.

The $oldsymbol{ ext{ extgamma, extdelta}}$ conjecture is equivalent to Dixmier's conjecture.

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 $\gamma,\delta$ 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 $A_1$ all the involutions belong to one conjugacy class, we show that every involutive endomorphism from $(A_1,\gamma)$ to $(A_1,\delta)$ is an automorphism ($\gamma$ and $\delta$ are two involutions on $A_1$), and given an endomorphism $f$ of $A_1$ (not necessarily an involutive endomorphism), if one of $f(X)$,$f(Y)$ is symmetric or skew-symmetric (with respect to any involution on $A_1$), then $f$ is an automorphism.