On Lie Algebras Consisting of Locally Nilpotent Derivations

Let K be an algebraically closed field of characteristic zero and A an integral K-domain. The Lie algebra Der(K)(A) of all K-derivations of A contains the set LND (A) of all locally nilpotent derivations. The structure of LND (A) is of great interest, and the question about properties of Lie algebras contained in LND (A) is still open. An answer to it in the finite dimensional case is given. It is proved that any subalgebra of finite dimension (over K) of Der(K)(A) consisting of locally nilpotent derivations is nilpotent. In the case A = K[x, y], it is also proved that any subalgebra of Der(K)(A) consisting of locally nilpotent derivations is conjugate by an automorphism of K[x, y] with a subalgebra of the triangular Lie algebra.