On the lifting of the Nagata automorphism

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, ..., 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.