On nilpotent and solvable Lie algebras of derivations

TL;DR

This paper investigates the structure of nilpotent and solvable Lie algebras of derivations over commutative algebras, establishing bounds on their derived length and characterizing low-rank cases, with implications for vector fields.

Contribution

It provides new bounds on the derived length of nilpotent and solvable Lie algebras of derivations and characterizes low-rank cases over the quotient field.

Findings

Derived length of nilpotent subalgebras is at most their rank over R.

Solvable Lie algebras have derived length at most twice their rank in characteristic zero.

Characterization of nilpotent and solvable Lie algebras of rank 1 and 2.

Abstract

Let K be a field and A be a commutative associative K-algebra which is an integral domain. The Lie algebra Der A of all K-derivations of A is an A-module in a natural way and if R is the quotient field of A, then RDer A is a vector space over R. It is proved that if L is a nilpotent subalgebra of RDer A 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 RDer A 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.