Solvable Lie algebras of derivations of polynomial rings in three variables

Abstract

Let K be an algebraically closed field of characteristic zero, 123 =[,,] Axxx K the polynomial ring in three variables and R = 123 (,,) xxx K the field of rational functions. If L is a subalgebra of the Lie algebra 3 () W K of all K-derivations of A , then RL is a Lie algebra over K and R dimRL will be called the rank of L over R. We study solvable subalgebras L of 3 () W K of rank 3 over R. It is proved that L is isomorphic to a subalgebra of the general affine Lie algebra 3 () aff K if L contains an abelian ideal I of rank 3 over R. If L has an ideal I with =2 R rkI , then L is contained in a subalgebra L of % 3 ()= WDerR K K such that L is an extension of a subalgebra of 2 () affF by a subalgebra of dimension ≤ 2 , where F is the field of constants of I in R .