ENDOTRIVIAL MODULES FOR NILPOTENT RESTRICTED LIE ALGEBRAS

DAVID J. BENSON AND JON F. CARLSON

Abstract. Let 𝔤 be a finite dimensional nilpotent p-restricted Lie algebra over a field k of characteristic p. For p ⩾ 5, we show that every endotrivial 𝔤-module is a direct sum of a syzygy of the trivial module and a projective module. The proof includes a theorem that the intersection of the maximal linear subspaces of the null cone of a nilpotent restricted p-Lie algebra for p ⩾ 5 has dimension at least two. We give an example to show that the statement about endotrivial modules is false in characteristic two. In characteristic three, another example shows that our proof fails, and we do not know a characterization of the endotrivial modules in this case.

1. Introduction

Let k be a field of characteristic p and G a finite group scheme. An endotrivial kG-module is a finitely generated module whose k-endomorphism ring is a trivial module in the stable category. The study of endotrivial modules for finite groups has a long history, beginning with the work of Dade [16, 17] and continuing with many other authors. A detailed history can be found in the survey papers [8, 24] and in the recent monograph of Mazza [23]. A primary motivation for the study comes from the fact that the group of endotrivial modules is an important part of the Picard group of self-equivalences of the stable category of finitely generated modules modulo projective modules.

The same definition of endotrivial modules and motivation for their study hold for more general finite group schemes. However, the results in the more general case, even for specific structures such as restricted p-Lie algebras, are not nearly as strong. For the group algebras of finite p-groups, there is a complete classification of the endotrivial modules [1, 6, 7, 12, 13, 14]. If G is a more general finite group, the structure of the group of endotrivial modules is discernable from the group structure, and the modules have been classified in many cases. For finite group schemes, it is known that there is only a finite number of isomorphism classes of endotrivial modules of any fixed dimension [11]. The paper [10] succeeds in giving a classification of endotrivial modules in the case that G is the finite group scheme of a restricted p-Lie algebra associated to a reductive algebraic group. The results also apply to the Lie algebras of some parabolic subgroups, including the Borel subgroup. However, for all of that, it is still not known if the group of endotrivial modules for a general finite group scheme, including that of a restricted p-Lie algebra, is finitely generated.

In this paper, we investigate endotrivial modules for nilpotent, finite dimesional restricted Lie algebras. Our main theorem reads as follows.

2010 Mathematics Subject Classification. 17B50.

The second author was partially supported by Simons Foundation Grant 054813-01.

Theorem 1.1. Let p ⩾ 5 and let 𝔤 be a nilpotent restricted Lie algebra over a field k of characteristic p. If M is an endotrivial module for 𝔤, then for some n ∈ ℤ we have that M ≅ Ω^{n}(k) ⊕ P where P is a projective module.

The proof of the theorem involves an examination of the restricted nullcone of 𝔤, leading to Theorem 3.2, which is actually a statement about the connectivity of the restricted null cone. Let {0} = Z_{0} ⊆ Z_{1} ⊆ Z_{2} ⊆ ⋯ be the upper p-central series for 𝔤. Let w = w(𝔤) be the maximum of the dimensions of Z_{i+1}/Z_{i}. Theorem 3.2 says that if k is perfect and its characteristic is p ⩾ 5 and w ⩾ 2, then the intersection of the irreducible components of the null cone of 𝔤 is a linear variety of dimension at least 2. In other words, in the case of restricted Lie algebras, large enough characteristic of the base field implies a lower bound on the dimension of the intersection of the components of the restricted null cone. Given the situation that we find here, it is tempting to conjecture that if k has characteristic p > 0, then the dimension of the intersection of the components of the restricted null cone of a nilpotent restricted Lie algebra 𝔤 over k has some lower bound, that increases as both p and w(𝔤) increase.

We show also that the proofs do not extend to restricted Lie algebras in characteristics p = 2 or p = 3. In the characteristic 2 case, the Lie algebra of strictly upper triangular 3 × 3 matrices over a field of characteristic 2 has the property that the intersection of the components of the restricted null cone has dimension one, and there actually exist endotrivial modules that are not isomorphic to Ω^{n}(k) for any n. We present an example of a null cone in characteristic 3, that fails to satisfy Theorem 3.2. It is possible that the algebra of that example satisfies the conclusion of our main theorem. However, a different proof would be needed to verify any such statement.

Throughout the paper, k denotes a field of prime characteristic p. All modules we consider are finitely generated. For M a module over a group algebra or a restricted Lie algebra, we write M ⊕ (proj) to mean the direct sum of M with some projective module, which may be the zero module.

Acknowledgment. Both authors would like to thank the University of Bielefeld for its hospitality while this work was carried out. The second author is grateful to the Mathematisches Forschungsinstitut Oberwolfach and to the Simons Foundation for financial support.

2. Restricted Lie algebras

In this section, we review some known results on restricted p-Lie algebras. We include some detail on such things as the definition, because the technicalities are needed.

We recall that a restricted Lie algebra 𝔤 over a field of characteristic p consists of a Lie algebra together with a p-power operation a ↦ a^{[p]} satisfying the following axioms:

(i) For all λ ∈ k, a ∈ 𝔤, we have (λa)^{[p]} = λ^{p}a^{[p]}. (ii) For all a, b ∈ 𝔤, we have [a^{[p]}, b] = Ad(a)^{p}(b). (iii) For all a, b ∈ 𝔤, we have (a + b)^{[p]} = a^{[p]} + b^{[p]} + ∑_{i=1}^{p−1} s_{i}(a, b) where (p − i)s_{i}(x, y)

is the coefficient of t^{p−i−1} in the formal expression Ad(tx + y)^{p−1}(x). In particular, s_{i}(a, b) is a linear combination of p-fold commutators of a and b.

In these axioms, for a ∈ 𝔤, Ad(a) denotes the derivation 𝔤 → 𝔤 given by Ad(a)(b) = [a, b]. Inductively, for n > 1 we write a^{[p^{n}]} for (a^{[p^{n−1}]})^{[p]}. We say that 𝔤 is cyclic if it is generated by a single element z. It is then the linear span of z, z^{[p]}, z^{[p^{2}]}, …, z^{[p^{n−1}]} with z^{[p^{n}]} = 0, for some n ⩾ 1. Note that a cyclic algebra is commutative.

Lemma 2.1. Let a and b be elements of a restricted Lie algebra 𝔤. Then (i) for all n ⩾ 1, we have [a^{[p^{n}]}, b] = Ad(a)^{p^{n}}(b), (ii) for all n ⩾ 1, a commutes with a^{[p^{n}]}, and (iii) if [a, b] commutes with b, then for all n ⩾ 1, a commutes with b^{[p^{n}]}.

Proof. (i) This follows from axiom (ii) above and induction on n.

(ii) By (i), we have [a^{[p^{n}]}, a] = Ad(a)^{p^{n}}(a) = Ad(a)^{p^{n}−1}[a, a] = 0.

(iii) Again using (i), we have [b^{[p^{n}]}, a] = Ad(b)^{p^{n}}(a) = Ad(b)^{p^{n}−2}[b, [b, a]] = 0. □

Definition 2.2. The upper p-central series of a p-restricted Lie algebra 𝔤 is defined as follows. We let Z_{1}(𝔤) be the subspace of the center Z(𝔤) consisting of the elements z with z^{[p]} = 0. For i ⩾ 2 we define Z_{i}(𝔤) to be the inverse image in 𝔤 of Z_{1}(𝔤/Z_{i−1}(𝔤)). Thus we have Z_{i}(𝔤)/Z_{i−1}(𝔤) = Z_{1}(𝔤/Z_{i−1}(𝔤)), and [𝔤, Z_{i}(𝔤)] ⊆ Z_{i−1}(𝔤).

If 𝔤 is nilpotent, then for i large enough we have 𝔤 = Z_{i}(𝔤). The quotient Lie algebras Z_{i}(𝔤)/Z_{i−1}(𝔤), are elementary, meaning that they are commutative and have trivial p-power operation.

Definition 2.3. If 𝔤 is a restricted Lie algebra, we write V (𝔤) for its restricted enveloping algebra. This is the quotient of the universal enveloping algebra U(𝔤) by the elements x^{p}−x^{[p]} for x ∈ 𝔤.

If x_{1}, …, x_{n} is a vector space basis for 𝔤, then the restricted version of the Poincaré– Birkhoff–Witt theorem states that V (𝔤) has a basis consisting of the monomials x_{1}^{a_{1}} ⋯ x_{n}^{a_{n}} where 0 ⩽ a_{i} < p for 1 ⩽ i ⩽ n. So V (𝔤) is an associative algebra of dimension p^{n} over k. It is a Frobenius algebra, so in particular it is self-injective.

A module for a restricted Lie algebra 𝔤 is a module for 𝔤 as a Lie algebra, with the property that the endomorphism representing x^{[p]} is the pth power of the endomorphism representing x. Thus modules for 𝔤 extend uniquely to modules for the associative algebra V (𝔤). For x ∈ 𝔤 and M a 𝔤-module, we write Im(x, M) = xM for the subspace of M that is the image of multiplication by x.

Lemma 2.4. Let 𝔤 be a nilpotent restricted Lie algebra. Then the norm element

N_{𝔤} = x_{1}^{p−1} ⋯ x_{n}^{p−1} ∈ V (𝔤)

is, up to scalar multiple, independent of choice of basis x_{1}, …, x_{n} of 𝔤, and spans the one dimensional socle of V (𝔤).

If M is a 𝔤-module then

(i) The module M has a non-trivial projective summand if and only if the image of N_{𝔤} on M is non-zero.

(ii) If w is a non-zero element of Z_{1}(𝔤) then Im(w^{p−1}, M) has a non-trivial projective summand as a 𝔤/(w)-module if and only if M has a non-trivial projective summand as a 𝔤-module.

Proof. The first statement and (i) are standard, so we prove (ii). The norm element for the one dimensional Lie algebra (w) is N_{(w)} = w^{p−1}. We may choose a basis x_{1}, …, x_{n} of 𝔤 with w = x_{n}, and then the images of x_{1}, …, x_{n−1} form a basis for 𝔤/(w). So if M is a 𝔤-module, then the image Im(w^{p−1}, M) of N_{(w)} on M is a 𝔤/(w)-module, and we have

Im(N_{𝔤/(w)}, Im(w^{p−1}, M)) = Im(N_{𝔤}, M).

Now apply (i). □

3. The restricted nullcone

Jantzen [22], Friedlander and Parshall [18] introduced the notion of the restricted null cone, and showed that it is the spectrum of the cohomology ring H^{∗}(𝔤, k). The aim of this section is to prove the main theorem on the connectness of the components of the null cone for larger characteristics.

Definition 3.1. The restricted nullcone N_{1}(𝔤) of a restricted Lie algebra 𝔤 is the set of y ∈ 𝔤 such that y^{[p]} = 0. This is a closed, conical subset of 𝔤. We write \overline{N}_{1}(𝔤) for the corresponding projective variety of one smaller dimension.

Theorem 3.2. Let 𝔤 be a finite dimensional nilpotent restricted Lie algebra over a field k of characteristic p. Assume that if p = 2 then k is quadratically closed, and that if p is odd then k is perfect.

(i) If 𝔤 is not cyclic then there exists a rank two elementary subalgebra 𝔢 ⊆ 𝔤, containing a non-zero element w ∈ Z_{1}(𝔤).

(ii) If p ⩾ 5, then 𝔢 can be chosen so that if y ∈ 𝔤 satisfies y^{[p]} = 0 then there is a restricted Lie subalgebra 𝔥 of 𝔤 containing 𝔢 and y with the property that N_{1}(𝔥) is equal to the linear span of 𝔢 and y.

Proof. (i) If Z_{1}(𝔤) is not cyclic, we may choose 𝔢 to be a subalgebra of Z_{1}(𝔤) of dimension two, and we are done. Otherwise, choose i minimal such that Z_{i+1}(𝔤) is not cyclic. Then Z_{i}(𝔤) is generated by an element z with the relation z^{[p^{i}]} = 0, and is spanned by z, z^{[p]}, …, z^{[p^{i−1}]}. In particular, Z_{i}(𝔤) is commutative.

We claim that the dimension of Z_{i+1}(𝔤)/Z_{i}(𝔤) is at least two. For suppose otherwise, and choose x ∈ Z_{i+1}(𝔤) \ Z_{i}(𝔤). Then x^{[p]} ∈ Z_{i−1}(𝔤) and [x, Z_{i+1}(𝔤)] = [x, Z_{i}(𝔤)] ⊆ Z_{i−1}(𝔤) and so x ∈ Z_{i}(𝔤), which is a contradiction.

Next, assume that p is odd. Choose x_{1}, x_{2} ∈ Z_{i+1}(𝔤) linearly independent modulo Z_{i}(𝔤). Since p is odd, the elements (x_{1}+λx_{2})^{[p]} and x_{1}^{[p]}+λ^{p}x_{2}^{[p]} are equivalent modulo Z_{i−1}(𝔤). Since Z_{i}(𝔤)/Z_{i−1}(𝔤) is one dimensional, and k is perfect, for some λ ∈ k the element (x_{1} + λx_{2})^{[p]} is in Z_{i−1}(𝔤). Thus we have shown that there is an element x ∈ Z_{i+1}(𝔤) \ Z_{i}(𝔤) such that x^{[p]} ∈ Z_{i−1}(𝔤). Because Z_{i}(𝔤) is cyclic, there is an element u ∈ Z_{i}(𝔤) with x^{[p]} = u^{[p]}. Now replacing x by x − u, again using the fact that p is odd, we have x^{[p]} ∈ Z_{i−2}(𝔤). Continuing this way and using induction, we eventually obtain an element x ∈ Z_{i+1}(𝔤) \ Z_{i}(𝔤) such that x^{[p]} = 0. Finally, we let 𝔢 be the span of x and w with w = z^{[p^{i−1}]}.

Now assume that p = 2. Let x_{1}, x_{2} ∈ Z_{i+1}(𝔤) be elements that are linearly independent modulo Z_{i}(𝔤). Then

(x_{1} + λx_{2})^{[2]} = x_{1}^{[2]} + λ[x_{1}, x_{2}] + λ^{2}x_{2}^{[2]}.

Because x_{1}^{[2]}, [x_{1}, x_{2}] and x_{2}^{[2]} are in Z_{i}(𝔤), and Z_{i}(𝔤)/Z_{i−1}(𝔤) is one dimensional, the fact that k is quadratically closed implies that for some λ ∈ k we have (x_{1} + λx_{2})^{[2]} ∈ Z_{i−1}(𝔤), and again we get an element x ∈ Z_{i+1}(𝔤) \ Z_{i}(𝔤) such that x^{[2]} ∈ Z_{i−1}(𝔤).

We now argue by downwards induction on j for 0 ⩽ j ⩽ i − 1 that we may rechoose x so that x^{[2]} ∈ Z_{j}(𝔤). We have dealt with the case j = i − 1. If this is true for j and if j ⩾ 1, we replace x by x + λz^{[2^{i−j−1}]}. We have x^{[2]}, [x, z^{[2^{i−j−1}]}] and z^{[2^{i−j}]} all in Z_{j}(𝔤), and Z_{j}(𝔤)/Z_{j−1}(𝔤) is one dimensional. So again using the fact that k is quadratically closed, there is a value of λ so that (x + λz^{[2^{i−j−1}]})^{[2]} is in Z_{j−1}(𝔤). Finally, taking j = 0, we have an element x ∈ Z_{i+1}(𝔤) \ Z_{i}(𝔤) such that x^{[2]} = 0. So we can let 𝔢 be the span of x and w with w = z^{[2^{i−1}]} and we are done also in the characteristic 2 case.

(ii) Now suppose that p ⩾ 5, and choose 𝔢 as in the proof of (i). Let y ∈ N_{1}(𝔤); in other words, y ∈ 𝔤 with y^{[p]} = 0. Then since x ∈ Z_{i+1}(𝔤), we have [x, y] ∈ Z_{i}(𝔤), and so [x, [x, y]] and [y, [x, y]] are in Z_{i−1}(𝔤). Write [x, [x, y]] = u^{[p]} and [y, [x, y]] = v^{[p]} with u, v ∈ Z_{i}(𝔤). Then [u, x], [u, y], [v, x] and [v, y] are in Z_{i−1}(𝔤). Since Z_{i}(𝔤) is commutative, we have [u, [u, x]] = 0, and hence [u^{[p]}, x] = Ad(u)^{p}(x) = 0. Similarly, [u^{[p]}, y] = 0, [v^{[p]}, x] = 0 and [v^{[p]}, y]=0. So [x, [x, y]] and [y, [x, y]] commute with x and y.

Thus any commutator having length at least four of x, y and elements of Z_{i}(𝔤) is zero. Let 𝔥 be the subalgebra spanned by x, y and Z_{i}(𝔤), so that 𝔥 has nilpotence class at most four. It follows, using axiom (iii) of the definition at the beginning of Section 2 of a restricted Lie algebra, that the p-power map is semilinear (via the Frobenius map) on 𝔥, since all the s_{j} involve commutators of length p, and we are assuming that p ⩾ 5. So for any λ, μ, ν ∈ k we have (λw + μx + νy)^{[p]} = 0 (recall that w = z^{[p^{i−1}]}), and N_{1}(𝔥) is the linear span in 𝔥 of w, x and y. □

4. Endotrivial modules

Recall that a 𝔤-module M is endotrivial if Hom_{k}(M, M) ≅ M ⊗ M^{∗} is isomorphic to k ⊕ (proj), the direct sum of a one dimensional trivial module and a projective module. For example, the modules Ω^{n}(k) are endotrivial. Here, for a module M, Ω(M) is the kernel of a projective cover φ : P → M (P a projective module, φ a surjective map). Dually, Ω^{−1}(M) is the cokernel of the injective hull M → I. Iterating, Ω^{i}(M) = Ω(Ω^{i−1}(M)) and Ω^{−i}(M) = Ω^{−1}(Ω^{−i+1}(M)).

Theorem 4.1. If 𝔤 is a nilpotent commutative restricted p-Lie algebra then every endotrivial module is isomorphic to Ω^{n}(k) ⊕ (proj) for some n. In addition, if 𝔤 is cyclic, then either M ≅ k ⊕ (proj) or M ≅ Ω(k) ⊕ (proj).

Proof. We observe that the restricted enveloping algebra of a nilpotent commutative restricted p-Lie algebra is isomorphic, as an algebra, to the group algebra of an abelian pgroup. It is shown in Theorem 5.6 of [9] that a module is endotrivial if and only if it has constant stable Jordan type [1] or [p−1]. This is independent of the Hopf structure. Finally, in an early paper on the subject Dade [17] showed that if G is an abelian p-group, then every indecomposable endotrival kG-module is isomorphic to Ω^{n}(k) for some n. The last statement is a consequence of the fact that if 𝔤 is cyclic, then Ω^{2}(k) ≅ k. □

We aim to show that the same holds in many (but not all) other cases, when 𝔤 is nilpotent. The main result of this section is the following proposition on which our method is based.

Proposition 4.2. Let 𝔤 be a nilpotent restricted Lie algebra, let w be a non-zero element of Z_{1}(𝔤), and let M be an endotrivial 𝔤-module. Suppose that there exists n ∈ ℤ such that for all extension fields K of k and all x ∈ N_{1}(𝔤_{K}) \ (w), the restriction of M_{K} to the two dimensional subalgebra 𝔢 of 𝔤_{K} spanned by w and x is isomorphic to Ω^{n}(K) ⊕ (proj). Then M is isomorphic to Ω^{n}(k) ⊕ (proj)

Proof. We may replace M by Ω^{−n}(M) and assume that M has no projective summands. Then our goal is to show that M ≅ k. Set M_{0} = Ker(w, M) and M_{1} = Im(w^{p−1}, M). Since M is endotrivial, we have M_{0}/M_{1} ≅ k, so we have a short exact sequence of 𝔤/(w)-modules

(4.3)  0 → M_{1} → M_{0} → k → 0.

Let K be an extension field of k, and let y be an element of N_{1}(𝔤/(w)). Then the inverse image 𝔥 of (y) in 𝔤 is commutative, and there are two possibilities for its structure. Either it is cyclic or elementary. If 𝔥 is cyclic then looking at the dimension of M modulo p, we see that the restriction of M to 𝔥 is isomorphic to K ⊕ (proj). If 𝔥 = 𝔢 is elementary, then by hypothesis, the restriction of M to 𝔢 is isomorphic to K ⊕ (proj). So in each case, if we take the sequence (4.3), tensor over k with K and restrict to 𝔥, it splits, and (M_{1})_{K}↓_{𝔥/(w)} is projective. It follows that the support variety of M_{1} as a module over 𝔤/(w) is empty. So by Theorem 8.1 of [2] (see also [19], [4]), it follows that M_{1} is projective as a module over 𝔤/(w). If M_{1} ≠ 0, then by Lemma 2.4 (ii), M has a projective summand. By assumption, this is not true, so we have M_{1} = 0 and M ≅ k. □

5. Vector bundles

For many restricted Lie algebras 𝔤, the restricted nullcone is a linear subspace of 𝔤. When this is the case, an endotrivial module over 𝔤 defines a vector bundle on projective space of the null cone. The bundle is an invariant of the endotrivial module, and can be used, as we do in the next section, in the analysis of the module. A main tool is the following theorem.

Theorem 5.1. For r ⩾ 2, every line bundle (invertible sheaf) on projective space ℙ^{r−1} is isomorphic to a twist O(ℓ) of the structure sheaf O for some ℓ ∈ ℤ, and if O(ℓ) ≅ O(ℓ′) then ℓ = ℓ′.

Proof. See Corollary II.6.17 in [21]. □

We recall from [20] the construction of vector bundles from modules of constant Jordan type, adapted to our situation (see also Section 8.4 of [3]). Let 𝔤 be a restricted Lie algebra such that the null cone N_{1}(𝔤) is a linear subspace of 𝔤. Let x_{1}, …, x_{r} be a basis for N_{1}(𝔤), and let k[y_{1}, …, y_{r}] = k[N_{1}(𝔤)] be the ring of polynomial functions on N_{1}(𝔤), with y_{i}(x_{j}) = δ_{i,j} (Kronecker delta). Then \overline{N}_{1}(𝔤) = Proj (k[N_{1}(𝔤)]) ≅ ℙ^{r−1}, projective space of dimension r − 1. Let O be the structure sheaf on \overline{N}_{1}(𝔤).

If M is a module for 𝔤, we write M̃ for the trivial vector bundle M ⊗_{k} O on \overline{N}_{1}(𝔤). The operator θ_{M}: M̃(j) → M̃(j + 1) is defined by

θ_{M}(m ⊗ f) = ∑_{i=1}^{r} x_{i}(m) ⊗ y_{i}f.

Here, the twist (j) depends on the context. The operator satisfies the condition (θ_{M})^{p} = 0, using any p consecutive values for the twist. Following [5], we define

F_{i}(M) = \frac{Ker(θ_{M}) ∩ Im(θ_{M}^{i−1})}{Ker(θ_{M}) ∩ Im(θ_{M}^{i})}

as a subquotient of M̃. For an endotrivial module M, there is only one F_{i} which takes non-zero value on M. Namely, one of F_{1}(M), F_{p−1}(M) is a line bundle and the remaining F_{i}(M) are zero.

For syzygies, we have

F_{p−i}(Ω(M)) ≅ F_{i}(M)(−p + i).

For a proof, see Theorem 3.2 of [5], where it is proved in the context of elementary abelian p-groups, but the same proof works in our context. Applying this formula twice, we see that

F_{i}(Ω^{2}(M)) ≅ F_{i}(M)(−p).

In particular, for the trivial module k we have that F_{1}(k) ≅ O, and we obtain

F_{1}(Ω^{2n}(k)) ≅ O(−pn).

Theorem 5.2. Let 𝔤 be a nilpotent restricted Lie algebra such that N_{1}(𝔤) is a linear subspace of 𝔤. If M is an indecomposable endotrivial 𝔤-module then M ≅ Ω^{n}(k) for some n ∈ ℤ.

Proof. First notice that we may assume that the field k is algebraically closed. That is, by the Noether–Deuring theorem (see [15, Theorem 29.7]), if M is a 𝔤-module and if K ⊗_{k} M ≅ Ω^{n}(K) as K ⊗ 𝔤-modules, for K the algebraic closure of k, then M ≅ Ω^{n}(k).

Our objective is to show, after replacing M by a suitable syzygy, if necessary, that M ≅ k. If N_{1}(𝔤) is one dimensional then by Theorem 3.2 (i), 𝔤 is cyclic. So by Theorem 4.1, either

M or Ω(M) is a trivial module, and we are done. On the other hand, if 𝔤 = 𝔢 is elementary of rank two, then again by Theorem 4.1, we are done.

So now we suppose that N_{1}(𝔤) has dimension at least two, and 𝔤 is not elementary of rank two. The restriction of M to a one dimensional restricted Lie subalgebra of 𝔤 is isomorphic to k ⊕ (proj) or Ω(k) ⊕ (proj). Replacing M by Ω(M) if necessary, we may assume that the dimension of M is congruent to one modulo p, so that the restriction to a one dimensional subalgebra is k ⊕ (proj).

If 𝔢 ⩽ 𝔤 is a two dimensional elementary subalgebra then the restriction of M to 𝔢 is an even syzygy of the trivial module. We claim that there exists a single value of n ∈ ℤ such that for all such choices of 𝔢, the restriction of M to 𝔢 is isomorphic to Ω^{2n}(k) ⊕ (proj). We regard 𝔢 as a two dimensional subspace of the vector space N_{1}(𝔤), and write 𝔢̄ for the corresponding copy of ℙ^{1} in the projective space \overline{N}_{1}(𝔤). Then we have

F_{1}(M)|𝔢̄ ≅ F_{1}(M↓_{𝔢}) ≅ F_{1}(Ω^{2n}(k)) ≅ O(−pn)

as line bundles on ℙ^{1}. By Theorem 5.1, every vector bundle on \overline{N}_{1}(𝔤) is isomorphic to some O(ℓ), and its restriction to ℙ^{1} is again O(ℓ). It follows that we have F_{1}(M) ≅ O(−pn). Hence the number n is determined by M, and is the same for all choices of 𝔢.

Finally, we apply Proposition 4.2 to deduce that M is isomorphic to Ω^{n}(k) for some n ∈ ℤ. □

6. The case p ⩾ 5

We are now in a position to prove our main theorem concerning endotrivial modules.

Theorem 6.1. Let p ⩾ 5 and let 𝔤 be a restricted Lie algebra over a field k of characteristic p. Then every indecomposable endotrivial module M for 𝔤 is isomorphic to Ω^{n}(k) for some n ∈ ℤ.

Proof. As in the beginning of the proof of Theorem 5.2, without loss of generality, we may assume that the field k is algebaically closed and hence perfect.

Choose an elementary subalgebra 𝔢 of 𝔤 of rank two, and w a non-zero element of Z_{1}(𝔤) contained in 𝔢, as in Theorem 3.2 (ii). If y ∈ N_{1}(𝔤) then there is a subalgebra 𝔥 containing 𝔢 and y such that N_{1}(𝔥) is the linear span of 𝔢 and y. By Theorem 5.2, the restriction of M to 𝔥 is isomorphic to Ω^{n}(k) ⊕ (proj), for some n ∈ ℤ. Restricting further to 𝔢, we see that the values of n are the same for all choices of y. We now apply Lemma 4.2 to deduce that M is isomorphic to Ω^{n}(k) ⊕ (proj). □

7. Examples for p = 2 and p = 3

The purpose of this section is to point out that the conclusion of Theorem 6.1 does not hold true if the requirement that p ⩾ 5 is removed. The characteristic two example below is well known, since its restricted enveloping algebra is isomorphic to the group algebra of a dihedral group of order eight. For this algebra, the conclusion of Theorem 6.1 is false. In the characteristic three example the conclusion of Theorem 3.2(ii) is not true. However, in this case, we have no characterization of the endotrivial modules.

If p = 2, let 𝔤 be the three dimensional Heisenberg restricted Lie algebra spanned by elements x, y and z satisfying [x, y] = z, [x, z] = 0, [y, z] = 0, and x^{[2]} = y^{[2]} = z^{[2]} = 0. This is the Lie algebra of upper triangular 3 × 3 matrices with zero diagonal. For this algebra, N_{1}(𝔤) is a union of two subspaces of dimension two. One is spanned by x and z, and the other is spanned by y and z. So 𝔤 does not satisfy the conclusion of Theorem 3.2 (ii). Moreover, there are endotrivial modules that are not isomorphic to Ω^{n}(k) for any n ∈ ℤ. For example, the natural three dimensional matrix representation M satisfies M ⊗ M^{∗} ≅ k ⊕ V (𝔤), so M is endotrivial. In fact, in this case the group of endotrivial modules is isomorphic to ℤ × ℤ, with basis Ω(k) and M.

If p = 3, let 𝔤 be the four dimensional restricted Lie algebra spanned by elements x, y, z and z^{[3]} satisfying [x, y] = z, [x, z] = z^{[3]}, [y, z] = z^{[3]}, [x, z^{[3]}] = 0, [y, z^{[3]}] = 0, [z, z^{[3]}] = 0, x^{[3]} = 0, y^{[3]} = 0 and z^{[9]} = 0. The null cone for this algebra is the set of all points (λ, μ, σ, τ) ∈ k^{4} such that

(λx + μy + σz + τz^{[3]})^{[3]} = λ^{3}x^{[3]} + λ^{2}μ[x, [x, y]] + λμ^{2}[y, [y, x]] + μ^{3}y^{[3]} + σ^{3}z^{[3]}
= (λμ(λ − μ) + σ^{3})z^{[3]}

is equal to zero. This is the hypersurface λμ(λ − μ) + σ^{3} = 0 in affine 4-space, and is an irreducible variety. However, it contains no linear subspace of dimension 3. Consequently, the conclusion of Theorem 3.2 is not satisfied by this algebra. It is not clear what the group of endotrivial modules for this algebra is.

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Institute of Mathematics, Fraser Noble Building, Univeristy of Aberdeen, Aberdeen AB24 3UE, United Kingdom

Department of Mathematics, University of Georgia, Athens GA 30602, USA