Journal of Lie Theory 27 (2017), No. 4, 1057--1068
Copyright Heldermann Verlag 2017
On Lie Algebras Consisting of Locally Nilpotent Derivations
Anatoliy Petravchuk
Dept. of Algebra and Mathematical Logic, Faculty of Mechanics and Mathematics, T. Shevchenko University, 64 Volodymyrska Street, Kyiv 01033, Ukraine
aptr@univ.kiev.ua
Kateryna Sysak
Dept. of Algebra and Mathematical Logic, Faculty of Mechanics and Mathematics, T. Shevchenko University, 64 Volodymyrska Street, Kyiv 01033, Ukraine
sysakkya@gmail.com
Let K be an algebraically closed field of characteristic zero and A an integral K-domain. The Lie algebra DerK(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 DerK(A) consisting of locally nilpotent derivations is nilpotent. In the case A = K[x, y], it is also proved that any subalgebra of DerK(A) consisting of locally nilpotent derivations is conjugate by an automorphism of K[x, y] with a subalgebra of the triangular Lie algebra.
Keywords: Lie algebra, vector field, triangular, locally nilpotent derivation.
MSC: 17B66, 17B05, 13N15
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