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Selecta Mathematica

, Volume 17, Issue 4, pp 935–945 | Cite as

On the lifting of the Nagata automorphism

  • Alexei Belov-Kanel
  • Jie-Tai Yu
Article

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 \({F\langle x,y,z\rangle}\). The proof is based on the following two new results, which have their own interests: degree estimate of \({{Q*_FF\langle x_1,\ldots,x_n\rangle}}\) and tameness of the automorphism group \({{{\rm Aut}_Q(Q*_FF\langle x,y\rangle)}}\). The structure of the group of all z-automorphisms of the free associative algebra \({F\langle x,y,z\rangle}\) over an arbitrary field F is also determined.

Keywords

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

Mathematics Subject Classification (2010)

Primary 13S10 16S10 Secondary 13F20 13W20 14R10 16W20 16Z05 

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Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Department of MathematicsBar-Ilan UniversityRamat-GanIsrael
  2. 2.Department of MathematicsThe University of Hong KongHong Kong SARChina

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