Automorphisms and derivations of free Poisson algebras in two variables

AbstractLet P be a free Poisson algebra in two variables over a field of characteristic zero. We prove that the automorphisms of P are tame and that the locally nilpotent derivations of P are triangulable

Free Akivis algebras, primitive elements, and hyperalgebras

Free Akivis algebras and primitive elements in their universal enveloping algebras are investigated. The conjecture of K .H . Hofmann and K. Strambach on the structure of primitive elements is proved to be not valid, and a full system of primitive elements in free nonassociative algebra is contructed

Polarization algebras and their relations

Using an approach to the Jacobian conjecture by L.M. Druzkowski and K. Rusek, G. Gorni and G. Zampieri, and A.V. Yagzhev, we describe a correspondence between finite dimensional symmetric algebras and homogeneous tuples of elements of polynomial algebras. We show that this correspondence closely relates Albert's problem in classical ring theory and the homogeneous dependence problem in affine algebraic geometry related to the Jacobian conjecture. We demonstrate these relations in concrete examples and formulate some open questions

The Nagata automorphism is wild

It is proved that the well known Nagata automorphism of the polynomial ring in three variables over a field of characteristic zero is wild, that is, it can not be decomposed into a product of elementary automorphisms

The strong Nagata conjecture

It is proved that there exist wild coordinates in the polynomial algebra in three variables over a field of characteristic zero. This result implies the famous Nagata conjecture

Generic, almost primitive and test elements of free Lie algebras

We construct a series of generic elements of free Lie algebras. New almost primitive and test elements were found. We present an example of an almost primitive element which is not generic.published_or_final_versio