On the lifting of the Nagata automorphism

Alexei Belov-Kanel, Jie Tai Yu

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4 Scopus citations


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〈x1, ..., xn〉 and tameness of the automorphism group Autq(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 languageEnglish
Pages (from-to)935-945
Number of pages11
JournalSelecta Mathematica, New Series
Issue number4
StatePublished - Dec 2011

Bibliographical note

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


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


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