On minimal non-elementary Lie algebras

The class of minimal non-elementary Lie algebras over a field F are studied. These are classified when F is algebraically closed and of characteristic different from 2,3. The solvable algebras in this class are also characterised over any perfect field.


Introduction
Groups and Lie algebras that just fail to have a particular property have been studied extensively in the hope of gaining some insight into just what makes the group or algebra have that property. They are also useful in constructing induction proofs. In group theory such groups are sometimes called critical.
A Lie algebra L is called elementary if the Frattini ideal, φ(S), of each of its subalgebras S, is trivial. It is called minimal non-elementary if each of its proper subalgebras is elementary but it is not elementary itself. These algebras were first studied by Towers in [5], and recently Stagg and Stitzinger ([3]) have classified such algebras when L 2 is nilpotent and the ground field is algebraically closed. In section 2 we extend this result first to any solvable Lie algebra over an algebraically closed field, and then show that there are no such non-solvable Lie algebras provided that the underlying field also has characteristic different from 2, 3. The final result characterises all minimal non-elementary solvable Lie algebras over a perfect field. An example is given to show that there are minimal non-elementary Lie algebras over the real field that are not of the type described in the result of Stagg and Stitzinger.
Throughout, L will denote a finite-dimensional Lie algebra over a field F . Algebra direct sums will be denoted by ⊕, whereas direct sums of the vector space structure alone will be denoted by+.

The results
We say that L is an E-algebra if φ(S) ⊆ φ(L) for every subalgebra S of L.   Proof. Suppose that L is minimal non-elementary and not solvable. Clearly L is an A-algebra and so has the form given in Theorem 2.3. Clearly R = 0, since, otherwise, L is elementary. Choose Thus, This holds for all 1 ≤ i ≤ n, and so L is elementary, by [4, Theorem 4.8], a contradiction. The result follows from Theorem 2.2.
The abelian socle, AsocL, of L is the sum of its minimal abelian ideals.
so C ∩ L 2 is an ideal of L, whence AsocL ∩ C = 0, a contradiction. It follows that 0 = φ(L) = AsocL is the biggest ideal of L properly contained in L 2 . Now suppose that dimB > 1 and let x ∈ B. Then C = L 2 + F x is elementary and so L 2 ⊆ N (C) = Asoc(C) by Theorem 7.4 of [4], and L 2 is completely reducible as an F x-module.
Then the minimum polynomial of the restriction of adx to C i is irreducible for each i, and so {(adx)| L 2 : x ∈ B} is a set of commuting semisimple operators. Let Ω be the algebraic closure of F and put L Ω = L ⊗ F Ω, and so on. Then φ(L Ω ) = φ(L) Ω , by [1]. Also, as F is perfect, {(adx)| L 2 Ω : x ∈ B Ω } and {(adx)| φ(L) Ω : x ∈ B Ω } are sets of simultaneously diagonalizable linear maps. Let f 1 , . . . , f s , c 1 , . . . , c t be a basis of these common eigenvectors, where is a maximal subalgebra of L Ω and φ(L) Ω ⊆ M , a contradiction. Hence dimB = 1 and we have (i).
If L is nilpotent then (ii) holds as in [7,Theorem 4.5]. The converse holds as in [7,Theorem 4.5] Note that any algebra satisfying the conditions specified in Theorem 2.5 (i) are Lie algebras. There are also algebras satisfying these conditions which have dimension greater than three, unlike those described in Theorem 2.2, as the following example shows. , all other products being zero. Then it is straightforward to check that L 2 = Re 2 + Re 3 + Re 4 + Re 5 is abelian, and that L has the unique minimal ideal Re 4 + Re 5 = φ(L). Over the complex field this is an elementary supersolvable Lie algebra.