Iena O. Centralizers of derivations and closed rational functions in two variables.

The dissertation is devoted to the investigation of the structure of some infinite dimensional Lie algebras. In the the first chapter we make a short review of the literature connected with the topics studied in the dissertation. In the second chapter we collect basic definitions and important facts that are used in the subsequent chapters. The third chapter is dedicated to closed rational functions, which are naturally connected with the description of the rings of constants for derivations of $k(x_1,...,x_n)$. We give a sufficient condition for a rational function $F=f/g$ to be closed, namely $F$ is closed if $f$ and $g$ are algebraically independent and at least one of them is irreducible. We also prove a criterion for being closed. A rational function $F=f/g$ is closed if and only if the pencil $af+bg$ contains only finitely many reducible hypersurfaces. Some sufficient conditions for a polynomial to be irreducible are also given. The fourth chapter deals with the Lie algebra $P_2(k)$ and with some similar algebras. In the case of zero characteristic of the ground field $k$ we give a complete description of the centralizers of elements and of the maximal abelian Lie subalgebras in the Lie (Poisson) algebra $P_2(k)$ of polynomials in two variables, as well as in the Lie algebras of rational functions and power series in two variables. This description is given in terms of closed polynomials (closed rational functions, maximal power series). We also describe the structure of the eigenspaces of inner derivations of $k[x, y]$ in the case of zero characteristic. In the case of positive characteristic $p=char(k)$ we prove a stronger version of Nowicki's result: centralizers of non-central elements from $P_2(k)$ are free modules of rank $p$ over the center $k[x^p, y^p]$. In the fifth chapter in the case of zero characteristic we characterize the centralizers of elements and the maximal abelian subalgebras in the Lie algebra $sa_2(k)$. This description is given in terms of closed and Jacobian polynomials. In particular centralizers of elements are either abelian subalgebras or solvable subalgebras of solvable length $2$ in $sa_2(k)$. Maximal abelian subalgebras in $sa_2(k)$ are either infinite dimensional or have dimension $2$. In positive characteristic we calculate the lower central series and the derived series of $sa_2(k)$. Some other results about the structure of $sa_2(k)$ have been obtained.