Ualbai Umirbaev
Wayne State University
Free Poisson fields and algebras.
Abstract: P. Cohn proved that free associative algebras are free ideal rings (FI-ring or fir), i.e., every submodule of a free module over a free associative algebra is free. This result easily implies that subalgebras of free Lie algebras are free and automorphisms of finitely generated free Lie algebras are tame.

Let P(x_1,...,x_n) be a free Poisson field, i.e., the quotient field of a free Poisson algebra with extended Poisson bracket. In the context of the mentioned above results we proved that
(a) the universal enveloping algebra P(x_1,...,x_n)^e of P(x_1,...,x_n) is a free ideal ring; and
(b) the automorphism group of a free Poisson field in two variables is isomorphic to the two dimentional Cremona group.

I will also give a survey of the latest results on free Poisson algebras.