Chief factors of Lie algebras

In group theory the chief factors allow a group to be studied by its representation theory on particularly natural irreducible modules. It is to be expected, therefore, that they will play an important role in the study of Lie algebras. In this article we survey a few of their properties.


Introduction
Throughout L will denote a finite-dimensional Lie algebra over a field F . We call a subalgebra I a subideal of a Lie algebra L if there is a chain of subalgebras I = I 0 < I 1 < . . . < I n = L, where I j is an ideal of I j+1 for each 0 ≤ j ≤ n − 1.
Put L 1 = L, L k+1 = [L k , L] for k ≥ 1. These are the terms of the lower central series for L. We say that L has nilpotency class n if L n = 0 but L n+1 = 0. Let U be a subalgebra of L. If F has characteristic p > 0 we call U nilregular if the nilradical of U , N (U ), has nilpotency class less than p − 1. If F has characteristic zero we regard every subalgebra of L as being nilregular. We say that U is characteristic in L if it is invariant under all derivations of L. Nilregular ideals of L have the property that their nilradicals are characteristic in L. Details of the results in this section can be found in [12]. This result was proved by Schenkman ( [6]) for fields of characteristic zero; in characteristic p it follows from a more recent result of Maksimenko ([4]). Similarly, we will call the subalgebra U solregular if the underlying field F has characteristic zero, or if it has characteristic p and the (solvable) radical of U , R(U ), has derived length less than log 2 p. Then we have the following corresponding theorem, which uses a result of Petravchuk ([5]).

Theorem 1.2.
(i) If I is a solregular ideal of L then R(I) ⊆ R(L). (ii) If I is a solregular subideal of L and every subideal of L containing I is solregular, then R(I) ⊆ R(L).
These enable us to determine what the minimal ideals of L look like. Theorem 1.3. Let L be a Lie algebra over a field F , and let I be a minimal non-abelian ideal of L. Then either (i) I is simple or (ii) F has characteristic p, N (I) has nilpotency class greater than or equal to p − 1, and R(I) has derived length greater than or equal to log 2 p.
As a result of the above we will call the subalgebra U regular if it is either nilregular or solregular; otherwise we say that it is irregular. Then we have the following corollary. Corollary 1.4. Let L be a Lie algebra over a field F . Then every minimal ideal of L is abelian, simple or irregular.
Block's Theorem on differentiably simple rings (see [2]) describes the irregular minimal ideals as follows.
Theorem 1.5. Let L be a Lie algebra over a field of characteristic p > 0 and let I be an irregular minimal ideal of L. Then I ∼ = S ⊗ O n , where S is simple and O n is the truncated polynomial algebra in n indeterminates. Moreover, N (I) has nilpotency class p − 1 and R(I) has derived length ⌈log 2 p⌉.

Primitive Lie algebras
Next we introduce the concept of a primitive Lie algebra. Details of the results in this section can be found in [10]. A word of warning -this terminology has been used for a different concept elsewhere. If U is a subalgebra of L we define U L , the core (with respect to L) of U to be the largest ideal of L contained in U . We say that U is core-free in L if U L = 0. We shall call L primitive if it has a core-free maximal subalgebra. The centraliser of U in L is C L (U ) = {x ∈ L : [x, U ] = 0}.
There are three types of primitive Lie algebra: 1. primitive of type 1 if it has a unique minimal ideal that is abelian; 2. primitive of type 2 if it has a unique minimal ideal that is non-abelian; and 3. primitive of type 3 if it has precisely two distinct minimal ideals each of which is non-abelian. Of course, primitive Lie algebras of types 2 and 3 are semisimple, and those of types 1 and 2 are monolithic. (A Lie algebra L is called monolithic if it has a unique minimal ideal W , the monolith of L.) Example 2.1. Examples of each type are easy to find.
1. Clearly every primitive solvable Lie algebra is of type 1. 2. Every simple Lie algebra is primitive of type 2.
3. If S is a simple Lie algebra then L = S ⊕ S is primitive of type 3 with core-free maximal subalgebra D = {s + s : s ∈ S}, the diagonal subalgebra of L.
Let M be a maximal subalgebra of L. Then M/M L is a core-free maximal subalgebra of L/M L . We say that M is 1. a maximal subalgebra of type 1 if L/M L is primitive of type 1; 2. a maximal subalgebra of type 2 if L/M L is primitive of type 2; and 3. a maximal subalgebra of type 3 if L/M L is primitive of type 3. We say that an ideal A is complemented in L if there is a subalgebra U of L such that L = A + U and A ∩ U = 0. For primitive solvable Lie algebras we have the following analogue of Galois' Theorem for groups.
1. If L is a solvable primitive Lie algebra then all core-free maximal subalgebras are conjugate. 2. If A is a self-centralising minimal ideal of a solvable Lie algebra L, then L is primitive, A is complemented in L, and all complements are conjugate.
The Frattini ideal of L, φ(L), is the core of intersection of the maximal subalgebras of L. We say that L is φ-free if φ(L) = 0. Then we have the following characterisation of primitive Lie algebras of type 1.
where W is the abelian monolith of L, C is an abelian subalgebra of L, every element of which acts semisimply on W , and S is a Levi subalgebra of L. 3. If L is solvable, then L is primitive if and only if it has a self-centralising minimal ideal A.
For type 2 we have

Chief factors
The factor algebra A/B is called a chief factor of L if B is an ideal of L and A/B is a minimal ideal of L/B. So chief factors are as described in Corollary 1.4 and Theorem 1.5. We can identify different types of chief factor; details for this section can be found in [10]. A chief factor A/B is called Frattini if A/B ⊆ φ (L/B) . This concept was first introduced in [8].
If there is a subalgebra, M such that L = A + M and B ⊆ A ∩ M, we say that A/B is a supplemented chief factor of L, and that M is a supplement of A/B in L. Also, if A/B is a non-Frattini chief factor of L, then A/B is supplemented by a maximal subalgebra M of L.
If A/B is a chief factor of L supplemented by a subalgebra M of L, and A ∩ M = B then we say that A/B is complemented chief factor of L, and M is a complement of A/B in L. When L is solvable, it is easy to see that a chief factor is Frattini if and only if it is not complemented. Then we have the following generalisation of the Jordan-Hölder Theorem.
be chief series for the Lie algebra L. Then there is a bijection between the chief factors of these two series such that corresponding factors are isomorphic as Lmodules and such that the Frattini chief factors in the two series correspond.
The number of Frattini chief factors or of chief factors which are complemented by a maximal subalgebra of a finite-dimensional Lie algebra L is the same in every chief series for L. However, this is not the case for the number of chief factors which are simply complemented in L; in [11] we determine the possible variation in that number.
Note that if L is a primitive Lie algebra of type 3, its two minimal ideals are not L-isomorphic, so we introduce the following concept. We say that two chief factors of L are L-connected if either they are L-isomorphic, or there exists an epimorphic image L of L which is primitive of type 3 and whose minimal ideals are L-isomorphic, respectively, to the given factors. (It is clear that, if two chief factors of L are L-connected and are not L-isomorphic, then they are nonabelian and there is a single epimorphic image of L which is primitive of type 3 and which connects them.) Then, as we would hope,   In other words, there are r ideals A 1 , . . . , A r of L such that Theorem 3.5. Let L be a solvable Lie algebra, and let C/R =C be the crown associated with a supplemented chief factor of L. ThenC is complemented inL, and any two complements are conjugate by an automorphism of the form 1 + ad a for some a ∈C.
Finally, in [1], Barnes determined for a solvable Lie algebra which irreducible L-modules A have the property that H 1 (L, A) = 0. Theorem 3.6. Let L be a solvable Lie algebra and let A be an irreducible L-module. Then H 1 (L, A) = 0 if and only if L has no complemented chief factor isomrphic to A.

Covering and Avoidance
The subalgebra U avoids the factor algebra We say that U has the covering and avoidance property of L if U either covers or avoids every chief factor of L. We also say that U is a CAP -subalgebra of L. Then these subalgebras give characterisations of solvable and supersolvable Lie algebras; details can be found in [9]. There It was shown in [8] that, when L is solvable, this definition does not depend on the choice of chief series, and that the prefrattini subalgebras of L cover the Frattini chief factors and avoid the rest; that is, they are CAP -subalgebras of L.
Further examples were given by Stitzinger in [7], where he proved the following result (see [7] for definitions of the terminology used).    Corollary 4.4. Let L be any solvable Lie algebra and let U be an ideally embedded subalgebra of L with K = N 2 (L) ⊆ U . Then U is a CAP -subalgebra of L.
Another set of examples of CAP -subalgebras, which don't require L to be solvable, is given by the next result.
Theorem 4.5. Let L be any Lie algebra, let U be a supplement to an ideal B in L, and suppose that B k ⊆ U for some k ∈ N. Then U is a CAP -subalgebra of L.
We can calculate the dimension of CAP -subalgebras in terms of the chief factors that they cover. Lemma 4.6. Let U be a CAP -subalgebra of L, let 0 = A 0 < A 1 < . . . < A n = L be a chief series for L and let I = {i : 1 ≤ i ≤ n, U covers A i /A i−1 }. Then dim U = i∈I (dim A i − dim A i−1 ).
We have the following characterisations of solvable and supersolvable Lie algebras.