On the lifting of the Nagata automorphism

TL;DR

This paper proves that the Nagata automorphism cannot be lifted to a z-automorphism of the free associative algebra, using new degree estimates and tameness results for automorphism groups.

Contribution

It introduces new degree estimates and tameness results that are crucial for understanding automorphism lifting problems in algebra.

Findings

Nagata automorphism cannot be lifted to a z-automorphism of free associative algebra

New degree estimate for ${Q*_FF<x_1,...,x_n>}$

Tameness of automorphism group ${ ext{Aut}_Q(Q*_FF<x,y>)}$

Abstract

It is proved that the Nagata automorphism (Nagata coordinates, respectively) of the polynomial algebra $F[x,y,z]$ over a field $F$ cannot be lifted to a $z$-automorphism ($z$-coordinate, respectively) of the free associative algebra $K<x,y,z>$. The proof is based on the following two new results which have their own interests: degree estimate of ${Q*_FF<x_1,...,x_n>}$ and tameness of the automorphism group ${\text{Aut}_Q(Q*_FF<x,y>)}$.