On nilpotent and solvable Lie algebras of derivations
Let K be a ﬁeld and A be a commutative associative K-algebra which is an integral domain. The Lie algebra DerA of all K-derivations of A is an A-module in a natural way and if R is the quotient ﬁeld of A then RDerA is a vector space over R. It is proved that if L is a nilpotent subalgebra of RDerA of rank k over R (i.e. such that dimR RL = 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 RDerA 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.