Sisak K. Locally nilpotent Lie algebras of derivations of commutative rings

The dissertation is devoted to studying locally nilpotent subalgebras of Lie algebras of derivations on associative commutative algebras over fields of characteristic zero. Let K be a field of characteristic zero and A an associative commutati- ve K-algebra that is an integral domain. The Lie algebra DerK A of all derivations of A is embedded into the Lie algebra W (A) spanned by all derivations of the form rD, where D DerK A and r belongs to the fracti- on field R over A. Each derivation D of A can be uniquely extended to a derivation of R, and if r ∈ R then one can define a derivation rD by setting rD(x) = r • D(x) for all x ∈ R. This dissertation deals with locally nilpotent subalgebras L of finite rank n (over R) of the Lie algebra W (A). The rank of a subalgebra L is defined as the dimension (over R) of the vector space RL spanned by all derivations rD for all r R and D L. If F = F (L) is the field of constants for L, then it is shown that F L is a locally nilpotent Lie algebra over F and obtained its structure in the cases when L is of rank 1, 2, and 3 over R. Besides, we consider nilpotent subalgebras L of rank n over R of W (A) such that the center Z(L) is of rank n 1 over R. It is proved that F L is contained in the Lie algebra of the particular form, which is pointed out in the paper.