On nilpotent and solvable Lie algebras of derivations

Abstract

Let K be a field and A be a commutative associative K-algebra which is an integral domain. The Lie algebra DerKA of all K-derivations of A is an A-module in a natural way, and if R is the quotient field of A then RDerKA is a vector space over R. It is proved that if L is a nilpotent subalgebra of RDerKA of rank k over R (i.e. such that dimRRL=k), then the derived length of L is at most k and L is finite dimensional over its field of constants. In case of solvable Lie algebras over a field of characteristic zero their derived length does not exceed 2k. Nilpotent and solvable Lie algebras of rank 1 and 2 (over R) from the Lie algebra RDerKA are characterized. As a consequence we obtain the same estimations for nilpotent and solvable Lie algebras of vector fields with polynomial, rational, or formal coefficients. Analogously, if X is an irreducible affine variety of dimension n over an algebraically closed field K of characteristic zero and AX is its coordinate ring, then all nilpotent (solvable) subalgebras of DerKAX have derived length at most n (2n respectively).