On nilpotent and solvable Lie algebras of derivations

Abstract

Abstract. 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 dim R 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.

Nilpotent Lie algebras of derivations with the center of small corank

“…Case 1. rk R Z = n − 1. Then FI is of codimension 1 in FL by Lemma 5 from [6]. Therefore dim F FI ≥ n because of dim F FL ≥ n + 1 and dim F FL/FI = 1.…”

“…Proof. By Lemma 4 from [6], I is an ideal of the Lie algebra L. Let us show that I is abelian. Let us choose an arbitrary basis D 1 , .…”

“…Then it is easy to see that FI 1 is an ideal of the Lie algebra FL of rank n − 1 over R and the Lie algebra FL/FI 1 is of dimension 1 over F (by Lemma 5 from [6]). Let us choose an arbitrary element D n ∈ FL \ FI 1 .…”

“…. , T s of the F-space FI (note that the Lie algebra FL is finite dimensional over the field F by [6]). Every basis element T i can be written in the form…”