A Lie algebra that can be written as a sum of two nilpotent subalgebras, is solvable

TL;DR

This paper proves that a finite-dimensional Lie algebra over a field with characteristic greater than 5, which can be expressed as a sum of two nilpotent subalgebras, must be solvable, extending known results to low characteristic cases.

Contribution

It establishes the solvability of such Lie algebras over fields of characteristic p>5, including low characteristic cases, using homological methods.

Findings

Lie algebra expressed as sum of two nilpotent subalgebras is solvable for p>5

Extension of results to low characteristic fields

Homological methods used in proof

Abstract

This is an old paper put here for archeological purposes. It is proved that a finite-dimensional Lie algebra over a field of characteristic p>5, that can be written as a vector space (not necessarily direct) sum of two nilpotent subalgebras, is solvable. The same result (but covering also the cases of low characteristics) was established independently by V. Panyukov (Russ. Math. Surv. 45 (1990), N4, 181-182), and the homological methods utilized in the proof were developed later in arXiv:math/0204004. Many inaccuracies in the English translation are corrected, otherwise the text is identical to the published version.