Український математичний конгрес - 2009

Юрій Боднарчук (Університет "Києво-Могилянська Академія", Київ, Україна)

Nagata's type automorphisms as the exponents of three root localy nilpotent derivations

Locally nilpotent derivations from a Lie algebra san of the special affine Cremona group are investigated in a connection with the root decomposition of san relative to the maximal standard torus. Earlier it was proved that all root locally nilpotent derivations are elementary ones. From this it immediately follows that a sum of two roots is locally nilpotent if it is a sum of two elementary derivations and has the triangular form only. Much interesting a class of three root localy nilpotent derivations because the well known Nagata's and Anick's exotic automorphisms of polynomial algebra can be obtained as the exponents of such derivations. We give full description of three root localy nilpotent derivations of polynomial algebras over a field of zero characteristic. With an application of the I. Shestakov's and Umirbaev's results we get a series of wild automorphisms of a polynomial algebra in three variables, which are the exponents of pointed derivations.