Outer restricted derivations of nilpotent restricted Lie algebras

In this paper we prove that every finite-dimensional nilpotent restricted Lie algebra over a field of prime characteristic has an outer restricted derivation whose square is zero unless the restricted Lie algebra is a torus or it is one-dimensional or it is isomorphic to the three-dimensional Heisenberg algebra in characteristic two as an ordinary Lie algebra. This result is the restricted analogue of a result of T\^og\^o on the existence of nilpotent outer derivations of ordinary nilpotent Lie algebras in arbitrary characteristic and the Lie-theoretic analogue of a classical group-theoretic result of Gasch\"utz on the existence of $p$-power automorphisms of $p$-groups. As a consequence we obtain that every finite-dimensional non-toral nilpotent restricted Lie algebra has an outer restricted derivation.


Introduction
In 1966 W. Gaschütz proved the following celebrated result: Theorem. (W. Gaschütz [3]) Every finite p-group of order > p has an outer automorphism of p-power order.
Since every finite nilpotent group is a direct product of its Sylow p-subgroups, the outer automorphism group of a finite nilpotent group is a direct product of the outer automorphism groups of its Sylow p-subgroups. Therefore it is a direct consequence of Gaschütz' theorem that every finite nilpotent group of order greater than 2 has an outer automorphism. This answers a question raised by E. Schenkman and F. Haimo in the affirmative (see [9]).
Since groups and Lie algebras often have structural properties in common, it seems rather natural to ask whether an analogue of Gaschütz' theorem holds in the setting of ordinary or restricted Lie algebras. In the case of ordinary Lie algebras a stronger version of such an analogue is already known and was established by S. Tôgô around the same time as Gaschütz proved his result.
Theorem. (S. Tôgô [11,Corollary 1]) Every nilpotent Lie algebra of finite dimension > 1 over an arbitrary field has an outer derivation whose square is zero.
In fact, Tôgô's result is more general (see [11,Theorem 1]) and is a refinement of a theorem of E. Schenkman that establishes the existence of outer derivations for non-zero finite-dimensional nilpotent Lie algebras (see [7,Theorem 4]). Much later the first author proved a restricted analogue of Schenkman's result for p-unipotent restricted Lie algebras (see [2,Corollary 5.2]). (Here we follow [1,Section I.4, Exercise 23, p. 97] by calling a restricted Lie algebra (L, [p]) p-unipotent if for every x ∈ L there exists some positive integer n such that x [p] n = 0.) In this paper we prove that every finite-dimensional nilpotent restricted Lie algebra L over a field F of characteristic p > 0 has an outer restricted derivation whose square is zero unless: (1) L is a torus, or (2) dim F L = 1, or (3) p = 2 and L is isomorphic to the three-dimensional Heisenberg algebra h 1 (F) over F as an ordinary Lie algebra (see Theorem 1). Indeed, in these three cases every nilpotent restricted derivation is inner. As a consequence we also obtain a generalization of [2, Corollary 5.2] to nontoral nilpotent restricted Lie algebras (see Theorem 2) which is the full analogue of Schenkman's result for nilpotent restricted Lie algebras.
In the following we briefly recall some of the notation that will be used in this paper. A derivation D of a restricted Lie algebra (L, [6,Section 4,(15), p. 21]) and the set of all restricted derivations of L is denoted by Der p (L). Observe that Der p (L) is a restricted Lie algebra (see [6,Theorem 4]) and that every inner derivation of L is restricted. In this paper we will use frequently without further explanation that the vector space of outer restricted derivations Der p (L)/ ad(L) of L is isomorphic to the first adjoint restricted cohomology space H 1 * (L, L) of L in the sense of Hochschild (see [4,Theorem 2.1]). More generally, we will need for any restricted L-module M the vector space    . Hencẽ D + B 1 * = 0, and therefore D + ad(L) = ι(D + B 1 * ) = 0, so that D is not inner. By using the injectivity of ι again, one deduces the following criterion for the existence of outer restricted derivations whose square is zero.
can be extended uniquely to a p-mapping [p] ′ : L → L making (L, [p] ′ ) a restricted Lie algebra. By construction, The ideal A -with trivial p-mapping -is also a p-ideal of (L, Remark. A similar proof shows that Proposition 1 holds more generally for abelian p-ideals A of L that do not necessarily contain the center of L if one replaces everywhere Z(L) by the image of A under the p-mapping of (L, [p]).
The next result shows that maximal abelian p-ideals of finite-dimensional nonabelian nilpotent restricted Lie algebras are self-centralizing and contain the center properly. Proof. Suppose by contradiction that C := Cent L (A) A. As C/A is a non-zero ideal of the nilpotent Lie algebra L/A, there is an element x in L that does not belong to A such that x + A ∈ Z(L/A) ∩ C/A. Put X := A + x p . Then X is an abelian p-ideal of L properly containing A which by the maximality of A implies that L = X is abelian contradicting the hypothesis that L is non-abelian.
It follows from A = Cent L (A) that A is a faithful L/A-module under the induced adjoint action. Suppose now that Z(L) = A. Then Z(L) is a faithful and trivial L/ Z(L)-module under the induced adjoint action which implies that L = Z(L) is abelian, which again is a contradiction.
Let T p (L) denote the set of all semisimple elements of a finite-dimensional nilpotent restricted Lie algebra L. Since L is nilpotent, we have that T p (L) ⊆ Z(L) and thus T p (L) is the unique maximal torus of L. Furthermore, T p (L) is a p-ideal of L and it follows from [10, Theorem 2.3.4] that L/I is p-unipotent for every p-ideal I of L that contains T p (L). These well-known results will be used in the following proofs without further explanation. Note that dim F A = dim F Z(L) · p dimF L/A shows again that A contains Z(L) properly if L is not abelian.

2.3.
Maximal p-ideals of nilpotent restricted Lie algebras. For the convenience of the reader we include a proof of the following result.
Lemma 3. Let L be a finite-dimensional non-toral nilpotent restricted Lie algebra over a field of characteristic p > 0. Then every maximal p-ideal of L containing the unique maximal torus of L has codimension one in L.
Proof. Let I be a maximal p-ideal of L that contains the unique maximal torus T p (L) of L. Then Z(L/I) = 0 and because L/I is p-unipotent, there exists a nonzero central element z of L/I such that z [p] = 0. Then Z := Fz is a one-dimensional p-ideal of L/I and the inclusion-preserving bijection between p-ideals of L/I and p-ideals of L that contain I in conjunction with the maximality of I yields that L/I = Z is one-dimensional.

Main results
The main goal of this paper is to establish the existence of nilpotent outer restricted derivations of finite-dimensional nilpotent restricted Lie algebras. It is well-known that the restricted cohomology of tori vanishes (see [4] and the main result of [5] or [2, Corollary 3.6]). Hence tori have no outer restricted derivations. For the convenience of the reader we include the following straightforward proof of the latter statement. p , we conclude that D(x) = 0 for every x ∈ L.
If L is one-dimensional, then either L is a torus or L = Fe with e [p] = 0. In the second case the (outer) restricted derivations of L coincide with the vector space endomorphisms End F (L) = F · id L of L and therefore no non-zero (outer) restricted derivation is nilpotent.
Indeed there is only one more finite-dimensional nilpotent restricted Lie algebra (up to isomorphism of ordinary Lie algebras) that has no nilpotent outer restricted derivations, namely the three-dimensional restricted Heisenberg algebra over a field of characteristic two. Since the derived subalgebra of any Heisenberg algebra is central, it follows from [10, Example 2, p. 72] that Heisenberg algebras are restrictable. For the proof of the next result one only needs that the image of every p-mapping is central so that the result does not depend on the particular choice of the p-mapping.

Proposition 5. Every nilpotent restricted derivation of a three-dimensional restricted Heisenberg algebra over a field of characteristic two is inner.
Proof. Let F be any field of characteristic 2 and let L be the three-dimensional Heisenberg algebra h 1 (F) = Fx + Fy + Fz defined by [x, y] = z ∈ Z(L). Since L ′ is central, we have that (2) L [2] ⊆ Z(L) = Fz . Proof. Assume that L is neither a torus nor one-dimensional. In the following T will denote the unique maximal torus T p (L) of L and by assumption L/T = 0. We proceed by a case-by-case analysis.
Case 1: There exists a maximal p-ideal I of L containing T such that Z(L) I. According to Lemma 3, I has codimension 1 in L and therefore L = Fx ⊕ I for any x ∈ Z(L)\I. Let 0 = z ∈ Z(I) and consider the linear transformation D of L defined by setting D(x) := z and D(y) := 0 for every y ∈ I. Then D is a derivation of L (see [1,Section I.4, Exercise 8(a), p. 92]) with D 2 = 0. Moreover, the inner derivation ad L a vanishes on x for every a ∈ L and thus D cannot be inner. Finally, as L/I is p-unipotent and I has codimension 1 in L, it follows that L [p] ⊆ I. This implies that D(l [p] ) = 0 = (ad L l) p−1 (D(l)) for every l ∈ L, and so D is restricted.
Note that Case 1 covers already the abelian case. So for the rest of the proof we may assume that L is not abelian.  Let E := Fe ⊕ A/A be the one-dimensional p-unipotent p-subalgebra generated by the image of e in L/A. Note that A ∩ J ⊆ A E and dim F A ∩ J = r · p d − 1. By construction, A is a free u(E)-module of rank r · p d−1 . Consequently, But this implies p d = 2 and r = 1. Hence F has characteristic 2 and dim F L = 3. Since the Heisenberg algebra is the only non-abelian nilpotent three-dimensional Lie algebra (up to isomorphism), this finishes the proof.
Theorem 1 in conjunction with the introductory remarks of this section yields a characterization of finite-dimensional nilpotent restricted Lie algebras in terms of the existence of nilpotent outer restricted derivations. Proof. Clearly, (ii) implies (iii) and it follows from Theorem 1 that (iii) implies (i). It remains to prove the implication (i)=⇒(ii) which is an immediate consequence of Proposition 4, the paragraph thereafter, and Proposition 5.
If we exclude the one-dimensional case and assume that the characteristic of the ground field is greater than two, then Theorem 1 can be used to characterize the tori among nilpotent restricted Lie algebras in terms of the non-existence of restricted derivations with various nilpotency properties.
Corollary 2. For a nilpotent restricted Lie algebra L of finite dimension > 1 over a field of characteristic p > 2 the following statements are equivalent: (i) L is a torus.
(ii) L has no non-zero restricted derivation.
(iii) L has no nilpotent outer restricted derivation.
(iv) L has no outer restricted derivation D with D 2 = 0.
Finally, we obtain from Theorem 1 the following generalization of [2, Corollary 5.2]: Theorem 2. Every finite-dimensional non-toral nilpotent restricted Lie algebra over a field of characteristic p > 0 has an outer restricted derivation.
Proof. According to Theorem 1 and the first paragraph after Proposition 4, it is enough to show the assertion for the three-dimensional restricted Heisenberg algebra L := h 1 (F) over a field F of characteristic 2. Denote the one-dimensional center of L by Z. If L is 2-unipotent, then the claim follows from [2,Corollary 5.2]. Suppose now that L is not 2-unipotent. Then the center Z of L is a one-dimensional torus. According to [2,Proposition 3.9] in conjunction with [2, Corollary 3.6], we obtain that H 1 * (L, L) ∼ = H 1 * (L/Z, L). Since L ′ is central, we have L [2] ⊆ Z, and consequently, L/Z is 2-unipotent. Suppose now that L has no outer restricted derivation and therefore H 1 * (L/Z, L) ∼ = H 1 * (L, L) = 0. Then an application of [2, Proposition 5.1] yields that L is a free u(L/Z)-module and it follows that 3 = dim F L ≥ dim F u(L/Z) = 4 , a contradiction.
Remark. It can also be seen directly that any three-dimensional restricted Heisenberg algebra h 1 (F) over a field F of characteristic 2 has an outer restricted derivation. Namely, let D be a linear transformation of h 1 (F) vanishing on its center Z(h 1 (F)) and inducing the identity transformation on h 1 (F)/ Z(h 1 (F)). Then it is straightforward to see that D is indeed an outer restricted derivation of h 1 (F).
Neither Theorem 1 nor Theorem 2 does generalize to non-toral solvable restricted Lie algebras with non-zero center as the direct product of a two-dimensional nonabelian restricted Lie algebra and a one-dimensional torus over any field of characteristic p > 0 shows. In this case it is not difficult to see that there are no outer restricted derivations.