## Abstract

It is proved that all wild z-automorphisms including the well-known Nagata automorphism (all wild z-coordinates including the Nagata coordinates, respectively) of the polynomial algebra F[x, y, z] over an arbitrary field F cannot be lifted to a z-automorphism (z-coordinate, respectively) of the free associative algebra 〈x, y, z〉. The proof is based on the following two new results, which have their own interests: degree estimate of Q *F F〈x_{1}, ..., x_{n}〉 and tameness of the automorphism group Aut_{q}(Q * F F〈x, y〉). The structure of the group of all z-automorphisms of the free associative algebra F〈x, y〉 over an arbitrary field F is also determined.

Original language | English |
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Pages (from-to) | 935-945 |

Number of pages | 11 |

Journal | Selecta Mathematica, New Series |

Volume | 17 |

Issue number | 4 |

DOIs | |

State | Published - Dec 2011 |

### Bibliographical note

Funding Information:The research of Jie-Tai Yu was partially supported by an RGC-GRF Grant.

## Keywords

- Automorphisms
- Canonical decomposation
- Coordinates
- Degree estimate
- Free associative algebras
- Lifting
- Nagata
- Polynomial algebras
- Sandwich
- Stable tameness
- Tame
- Wild