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

Abstract

In 1963 O. Kegel raised the following question: is a Lie ring written as a sum of two nilpotent subrings solvable? Recently, Kostrikin [1] brought a renewed attention to this question in the case of finite-dimensional algebras over a field. The question is easily solved in the affirmative in the case of characteristic zero (cf.[1] or [2]). The purpose of this note is to prove the following theorem.Theorem. Over a field of characteristic p > 5, a finite-dimensional Lie algebra written as a sum of two nilpotent subalgebras, is solvable., a similar statement is proved under additional restrictions on the nilpotency index of one summand (with fewer restrictions on the characteristic of the ground field). Note that the theorem is no longer true when p = 2 (an appropriate counter example has been constructed in [4]). We make an essential use of Weisfeiler's results [6] on Lie algebras with a solvable maximal subalgebra which dictates a restriction upon the characteristic of the ground field.We turn to the proof of the theorem. We may assume the ground field K to be algebraically closed. Let L be a counter example to the theorem having the least possible dimension, let L = A + B be the corresponding decomposition into a sum of nilpotent subalgebras, where dim A ≤ dim B. The following lemma can be easily deduced from the minimality of the counter example (cf.[1])., then L 0 is solvable. In particular, L possesses a solvable maximal subalgebra.The following result has been proved in [6]: a maximal solvable subalgebra in a simple Lie algebra determines a long filtration in it. A closer analysis of this proof enables us to replace the simplicity condition by semisimplicity. Indeed, a semisimple Lie algebra with a solvable maximal subalgebra possesses a unique ideal by Block's theorem [7]. The proof of Theorem 1.2.2 in [6] may be repeated for such algebras almost verbatim. But the proof of Theorem 1.5.1 relies only on the conclusion of Theorem 1.2.2 and never makes use of the simplicity of the algebra.Let L 0 be a maximal solvable subalgebra of L containing A (it exists due to Lemma 1). By the above, we can apply Theorem 2.1.3 in [6] which states that for the filtrationby the subalgebra L 0 , the associated graded algebra will have the form grL = G = S ⊗ O m + D, where S is a simple Lie algebra isomorphic to sl 2 (K) or to the Zassenhaus algebra W 1 (n), O m is the algebra of truncated polynomials in m variables, D is a solvable subalgebra of W m = Der(O m ) such that O m contains no D-invariant ideals. The grading is given as follows:where S = i≥−1 e i is the standard grading. Henceforth, this particular grading is meant each time we refer to S ⊗ O m + D as a graded algebra.

Derived length and nildecomposable Lie algebras

“…In characteristic p the situation is as usual more complicated. A nildecomposable Lie algebra is solvable over a field of characteristic p with p ≥ 5, see [25] and the references therein. There is a counterexample in characteristic 2, see [21].…”