COHOMOLOGY OF UNIFORMLY POWERFUL p-GROUPS

WILLIAM BROWDER AND JONATHAN PAKIANATHAN

Abstract. In this paper we will study the cohomology of a family of p-groups associated to 𝔽_{p}-Lie algebras. More precisely we study a category BGrp of p-groups which will be equivalent to the category of 𝔽_{p}-bracket algebras (Lie algebras minus the Jacobi identity). We then show that for a group G in this category, its 𝔽_{p}-cohomology is that of an elementary abelian p-group if and only if it is associated to a Lie algebra.

We then proceed to study the exponent of H^{∗}(G; ℤ) in the case that G is associated to a Lie algebra 𝔏. To do this, we use the Bockstein spectral sequence and derive a formula that gives B_{2}^{∗} in terms of the Lie algebra cohomologies of 𝔏. We then expand some of these results to a wider category of p-groups. In particular we calculate the cohomology of the p-groups Γ_{n,k} which are defined to be the kernel of the mod p reduction GL_{n}(ℤ/p^{k+1}ℤ) ⟶^{mod} GL_{n}(𝔽_{p}). 1991 Mathematics Subject Classification. Primary: 20J06, 17B50; Secondary: 17B56.

1. Introduction and Motivation

Throughout this paper, p will be an odd prime. First some definitions:

Definition 1.1. Given a p-group G, Ω_{1}(G) = ⟨g ∈ G : g^{p} = 1⟩ where the brackets mean “smallest subgroup generated by”. G is called p-central if Ω_{1}(G) is central.

Definition 1.2. For any positive integer k, G^{p^{k}} = ⟨g^{p^{k}} : g ∈ G⟩.

Definition 1.3. F rat(G) = G^{p}[G, G].

We will study a category BGrp of p-groups which is naturally equivalent to the category of bracket algebras over 𝔽_{p}. This is exactly the category of p-central, pgroups G which have Ω_{1}(G) = G^{p} = F rat(G). For such a group G, the associated bracket algebra will be called Log(G).

In the case that one of these groups is associated to a Lie algebra, we will show that it has the same 𝔽_{p}-cohomology as an elementary abelian p-group. More precisely we will show: (The necessary definitions for the theorems quoted in this introductory section can be found in the relevant parts of the paper.)

Theorem 1 ( 2.10). Let G ∈ Obj(BGrp) and n = dim(Ω_{1}(G)). Then

H^{∗}(G; 𝔽_{p}) = ∧(x_{1}, …, x_{n}) ⊗ 𝔽_{p}[s_{1}, …, s_{n}]

(where the x_{i} have degree 1 and the s_{i} have degree 2) if and only if Log(G) is a Lie algebra. When this is the case, the polynomial algebra part restricts isomorphically to that of H^{∗}(Ω_{1}(G); 𝔽_{p}) and the exterior algebra part is induced isomorphically from that of H^{∗}(G/Ω_{1}(G); 𝔽_{p}) via the projection homomorphism.

(This theorem was proven independently by T. Weigel for the case p ≠ 3 in [W1], [W2].)

However the integral cohomology of these groups is complicated and indeed we will show that one can recover the Lie algebra associated to the group from knowledge of the Bockstein on its 𝔽_{p}-cohomology. To do this we calculate the full structure of the 𝔽_{p}-cohomology as a Steenrod-module. As an application, in corollary 2.26, we completely determine the comodule algebra structure (see [W2]) of H^{∗}(G; 𝔽_{p}).

With the mild additional hypothesis that the associated Lie algebra lifts to one over ℤ/p^{2}ℤ, we compute B_{2}^{∗} of the Bockstein spectral sequence in terms of the Lie algebra cohomologies of the corresponding Lie algebra:

Theorem 2 ( 2.36). Let G ∈ Obj(BGrp) with Log(G) = 𝔏, a Lie algebra, and suppose that 𝔏 lifts to a Lie algebra over ℤ/p^{2}ℤ. Then B_{2}^{∗} of the Bockstein spectral sequence for G is given by

B_{2}^{∗} = ⊕_{k=0}^{∞}H^{∗−2k}(𝔏; S^{k}).

Here S^{k} is the 𝔏-module of symmetric k-forms on 𝔏 with the usual action. (This action is described in detail before the proof of the theorem.)

The integral cohomology of p-groups is a rich subject and there are many theorems and conjectures about the exponent of H̄^{∗}(P; ℤ). (Where the bar denotes reduced cohomology and we recall that the exponent of an abelian group G is the smallest positive integer n such that ng = 0 for all g ∈ G). We will obtain some partial results on the exponent of the integral cohomology of the p-groups studied in this paper via the Bockstein spectral sequence.

In the last section of the paper, we extend the results mentioned to a bigger family of groups: the uniform, p-central, p-groups. (Elementary abelian p-groups are uniform and more generally one defines inductively that a p-central, p-group G is uniform if and only if Ω_{1}(G) = G^{p^{k}} for some nonnegative integer k and G/Ω_{1}(G) is itself uniform. Thus a uniform p-group will give rise to a tower of uniform pgroups called a uniform tower where each group in the tower is the quotient of the previous group G by Ω_{1}(G). In this paper such uniform towers will be indexed so that G_{1} is always an elementary abelian p-group. When this is done, G_{2} will always correspond to a group in the category BGrp.)

Specifically we prove:

Theorem 3 ( 3.14). Fix p ≥ 5. Let

G_{k+1} → G_{k} → ⋯ → G_{2} → G_{1} → 1

be a uniform tower with k ≥ 2. Let 𝔏 = Log(G_{2}) and let c_{ij}^{k} be the structure constants of 𝔏 with respect to some basis. Then for suitable choices of degree 1 elements x_{1}, …, x_{n} and degree 2 elements s_{1}, …, s_{n}, one has

H^{∗}(G_{k}; 𝔽_{p}) ≅ ∧^{∗}(x_{1}, …, x_{n}) ⊗ 𝔽_{p}[s_{1}, …, s_{n}]

with

\begin{matrix}β(x_{t}) = & − ∑_{i<j}^{n} c_{ij}^{t}x_{i}x_{j} \\ β(s_{t}) = & ∑_{i,j=1}^{n} c_{ij}^{t}s_{i}x_{j} + η_{t} \end{matrix}

for all t = 1, … n. Furthermore the η_{t} define a cohomology class [η] ∈ H^{3}(𝔏; ad) which vanishes if and only if there exists a uniform tower

G′_{k+2} → G′_{k+1} → G_{k} → ⋯ → G_{2} → G_{1} → 1

or in other words if and only if Log(G_{k}) (which is a Lie algebra over ℤ/p^{k−1}ℤ), has a lift to a Lie algebra over ℤ/p^{k}ℤ. Thus in this case where the Lie algebra lifts, one can drop the η_{t} term in the formula for the Bockstein.

A version of this theorem for the category of powerful, p-central, p-groups is also stated in theorem 3.16.

Some examples of p-groups covered in this paper are the groups Γ_{n,k} for n, k ≥ 1, where the group Γ_{n,k} is defined as the kernel of the map

GL_{n}(ℤ/p^{k+1}ℤ) ⟶^{mod} GL_{n}(𝔽_{p}).

2. Bracket groups

2.1. Some preliminaries. First we will recall a method of obtaining a bracket algebra from a p-group under suitable conditions.

Let us consider the following situation. Suppose we have a central short exact sequence of finite groups:

1 → V → G →^{π} W → 1

where W and V are elementary abelian p-groups, then we define

⟨·, ·⟩ : W × W → V given by ⟨x, y⟩ = x̂ŷx̂^{−1}ŷ^{−1} (1)

where x̂, ŷ are lifts of x, y to G, i.e., π(x̂) = x, π(ŷ) = y.

We also define a p-power map function

ϕ : W → V given by ϕ(x) = x̂^{p}. (2)

It is routine to verify that ⟨·, ·⟩ is a well-defined alternating, bilinear map and that ϕ is a well-defined linear map. (the linearity of ϕ uses that p is odd.)

We will now restrict ourselves to the case where the p-power map ϕ is an isomorphism. So V and W are isomorphic. In this case one can define

[·, ·] : W ⊗ W → W by [w_{1}, w_{2}] = \frac{1}{2}ϕ^{−1}(⟨w_{1}, w_{2}⟩). (3)

Then it is easy to see this is still an alternating, bilinear map on W. Note W is a vector space over 𝔽_{p}, the field on p elements. (The factor of \frac{1}{2} is put in for convenience in order to avoid messy expressions later on.)

Definition 2.1. A bracket algebra over 𝔽_{p} is a finite dimensional vector space W over 𝔽_{p} equipped with an alternating, bilinear form [·, ·] : W ⊗ W → W. The dimension of a bracket algebra is the dimension of the underlying vector space.

Definition 2.2. Brak is the category whose objects consist of bracket algebras over 𝔽_{p} and with morphisms the linear maps which preserve the brackets.

Now we can define a category of p-groups which is essentially equivalent to the category Brak. These groups will be our primary objects of study in the first part of this paper.

Definition 2.3. A p-power exact extension is a central short exact sequence:

1 → V → G →^{π} W → 1

where V, W are elementary abelian and where the p-power map ϕ is an isomorphism.

Note in such an extension, V = Ω_{1}(G) and W = G/Ω_{1}(G) (see definition 1.1). So any homomorphism between two groups G_{1}, G_{2} in the middle of p-power exact extensions takes V_{1} = Ω_{1}(G_{1}) into V_{2} = Ω_{1}(G_{2}) and hence induces maps V_{1} → V_{2} and W_{1} → W_{2} where W_{i} = G_{i}/Ω_{1}(G_{i}). These will be called respectively the induced maps on the V, (W) level.

Definition 2.4. BGrp is the category whose objects consist of finite groups G which fit in the middle of a p-power exact extension. These will be called bracket groups. The morphisms in BGrp are just the usual group morphisms between these groups, except we will identify two morphisms if they induce the same map on the V and W levels.

One can show that the two categories Brak and BGrp are naturally equivalent (see [Pak]). This follows from a general exponent-log correspondence (see [W1],

[W3] or [Laz]), suitably reworded for our purposes. We will give a short description of the functors involved in this equivalence of categories.

Define a covariant functor Log : BGrp → Brak as follows, to a bracket group G we associate the bracket algebra Log(G) = (W, [·, ·]) which is obtained as explained before. Notice that the underlying vector space of Log(G) is just G/Ω_{1}(G) so given ψ ∈ Mor(G_{1}, G_{2}) we note ψ induces a well-defined linear map Log(ψ) : Log(G_{1}) → Log(G_{2}), and it is easy to check that this is a map of bracket algebras.

A description of the inverse functor Exp : Brak → BGrp is given as follows. Given (L, [·, ·]) a bracket algebra, let K be a free ℤ/p^{2}ℤ-module of rank equal to the dimension of L. Let ϕ : L → K and π : K → L be injective/surjective maps such that ϕ ∘ π = p (multiplication by p). Then Exp(L) = (K, ∘) where

l ∘ m = l + m + ϕ[π(l), π(m)]. (4)

(Here + is the addition of K as a ℤ/p^{2}ℤ-module. Notice in this notation, 0 is the identity of Exp(L), l^{−1} corresponds to −l, l^{p} corresponds to pl, and the p-power map is indeed ϕ. )

For clarity we state the following proposition which summarizes these facts:

Proposition 2.5. The functors Log : BGrp → Brak and Exp : Brak → BGrp give a natural equivalence between the categories BGrp and Brak. Thus to every bracket algebra there naturally corresponds a unique bracket group.

Example:

A direct computation shows that the group Γ_{n,2} which is the kernel of the map

GL_{n}(ℤ/p^{3}ℤ) ⟶^{mod} GL_{n}(𝔽_{p}),

is a bracket group, and that Log(Γ_{n,2}) = 𝔤𝔩_{n}, the Lie algebra of n × n matrices.

Now we will study the groups in BGrp from a cohomological viewpoint. Recall if G ∈ Obj(BGrp) then G fits in a central short exact sequence

1 → V → G →^{π} W → 1

where we can identify V and W via the p-power isomorphism ϕ. If we do not put the restriction that ϕ is an isomorphism then such extensions as above are in one to one correspondence with H^{2}(W; V). Now for W an elementary abelian p-group of rank n (recall p is odd), we have that

H^{∗}(W; 𝔽_{p}) ≅ ∧^{∗}(x_{1}, …, x_{n}) ⊗ 𝔽_{p}[βx_{1}, …, βx_{n}]

the tensor product of an exterior algebra on degree 1 generators and a polynomial algebra on degree 2 generators which are the Bocksteins of the degree 1 generators. (Here β : H^{1}(W; 𝔽_{p}) → H^{2}(W; 𝔽_{p}) is the Bockstein. ) In basis free notation, this can be written

H^{∗}(W; 𝔽_{p}) ≅ ∧^{∗}(W^{∗}) ⊗ 𝔽_{p}[β(W^{∗})]

where W^{∗} = H^{1}(W; 𝔽_{p}) is the dual vector space of W and β(W^{∗}) is its image under the Bockstein β. Now,

H^{2}(W; V) ≅ H^{2}(W; 𝔽_{p}) ⊗ V by the Universal Coefficient Theorem
≅ (H^{2}(W; 𝔽_{p}))^{n} using a choice of basis of V.

(5)

Note that a bracket on W gives us a map br : ∧^{2}(W) → W. If we take the dual of this map we get a map br^{∗} : W^{∗} → ∧^{2}(W^{∗}). (Once we identify the dual of ∧^{2}(W) with ∧^{2}(W^{∗}) in the usual way.) With this notation, it is easy to argue by comparisons (see [BC]) and from the description of the Exp functor in equation 4 that the extension elements are of the form β(x_{i}) + br^{∗}(x_{i}) for i = 1, …, n. (If we use a suitable basis for V in equation 5. One chooses a basis E for W, the basis ϕ(E) for V and the canonical dual basis for V^{∗} and W^{∗} so for example {x_{1}, …, x_{n}} are dual to E.) In this formulation br^{∗}(x_{i}) are just the components of the bracket of Log(G), with respect to the basis E. We will use this form of the extension element from now on.

2.2. Lie Algebras and co-Lie algebras.

Definition 2.6. Given a bracket algebra (W, [·, ·]) we define the Jacobi form of (W, [·, ·]) to be

J(x, y, z) = ([[x, y], z] + [[y, z], x] + [[z, x], y]).

This is easily checked to be an alternating 3-form. Thus J : ∧^{3}(W) → W. Bracket algebras with J = 0 are called Lie algebras.

Given a bracket algebra, (W, [·, ·]), the bracket defines a map br : ∧^{2}(W) → W. Thus by taking duals and identifying the dual of ∧^{∗}(W) with ∧^{∗}(W^{∗}) in the usual way, we obtain a map br^{∗} : W^{∗} → ∧^{2}(W^{∗}), where W^{∗} is the dual vector space of W. This motivates the following definition:

Definition 2.7. A co-bracket algebra is a vector space W^{∗} equipped with a linear map br^{∗} : W^{∗} → ∧^{2}(W^{∗}). We will always extend br^{∗} as a degree 1 map on the whole of the graded algebra ∧^{∗}(W^{∗}) in the unique way that makes it a derivation on that algebra. We call such a co-bracket algebra a co-Lie algebra if br^{∗} ∘ br^{∗} = 0.

As was mentioned before, the dual of a bracket algebra, has a natural co-bracket algebra structure and similarly, the dual of a co-bracket algebra has a natural bracket algebra structure.

Given a co-bracket algebra W^{∗}, it is easy to check, as br^{∗} is a derivation, that br^{∗} ∘ br^{∗} = 0 if and only if br^{∗} ∘ br^{∗} : W^{∗} = ∧^{1}(W^{∗}) → ∧^{3}(W^{∗}) is zero. The dual of this map is a map from ∧^{3}(W) → W (where we are calling W^{∗∗} = W). It is easy to check that this is just the Jacobi form defined before for the bracket algebra W. Thus we can conclude that the dual of a co-Lie algebra is a Lie algebra and vice versa.

Let π : ∧^{2}(W) → W ⊗ W be the map defined by π(a ∧ b) = a ⊗ b − b ⊗ a. Thus the usual definition of a module over a bracket (Lie) algebra can be worded in the following more categorical way:

Definition 2.8. We say V is a module over the bracket algebra (W, br) if we have a map λ : W ⊗ V → V such that the following diagram commutes:

W ∧ W ⊗ V ⟶_{br⊗1} W ⊗ V
(1⊗λ)∘(π⊗1)↓  ↓^{λ}
W ⊗ V ⟶^{λ} V

Taking the dual of this definition, and noting that the dual of π, π^{∗} : W^{∗}⊗W^{∗} → ∧^{2}(W^{∗}) is just the canonical quotient map, gives us the following definition:

Definition 2.9. We say V^{∗} is a comodule over the co-bracket algebra (W^{∗}, br^{∗}) if we have a map λ^{∗} : V^{∗} → W^{∗} ⊗ V^{∗} such that the following diagram commutes:

W^{∗} ∧ W^{∗} ⊗ V^{∗} ⟵_{br^{∗}⊗1} W^{∗} ⊗ V^{∗}
1∧λ^{∗}↑  ↑_{λ^{∗}}
W^{∗} ⊗ V^{∗}  ⟵^{λ^{∗}} V^{∗}

It is easy to see that the dual of a comodule over the co-bracket algebra W^{∗} is a module over the bracket algebra W = W^{∗∗} and vice-versa.

2.3. The 𝔽_{p}-cohomology of a group in BGrp. In this section we will show that a group in BGrp has the 𝔽_{p}-cohomology of an elementary abelian p-group if and only if it is associated to a Lie algebra. More precisely we will prove:

Theorem 2.10. Let G ∈ Obj(BGrp) and n = dim(Ω_{1}(G)). Then

H^{∗}(G; 𝔽_{p}) = ∧(x_{1}, …, x_{n}) ⊗ 𝔽_{p}[s_{1}, …, s_{n}]

(where the x_{i} have degree 1 and the s_{i} have degree 2) if and only if Log(G) is a Lie algebra. When this is the case, the polynomial algebra part restricts isomorphically to that of H^{∗}(Ω_{1}(G); 𝔽_{p}) and the exterior algebra part is induced isomorphically from that of H^{∗}(G/Ω_{1}(G); 𝔽_{p}) via the projection homomorphism.

Fix G ∈ Obj(BGrp) we now study the Lyndon-Hochschild-Serre (L.H.S.) spectral sequence (with 𝔽_{p}-coefficients) associated to the extension

1 → Ω_{1}(G) = V →^{i} G →^{π} W → 1.

Recall that this is a spectral sequence with

E_{2}^{p,q} = H^{p}(W, H^{q}(V, 𝔽_{p}))

and it abuts to H^{∗}(G; 𝔽_{p}). Since this is a central extension, if we use 𝔽_{p} coefficients throughout then

E_{2}^{p,q} = H^{p}(W, H^{q}(V, 𝔽_{p})) = H^{q}(V, 𝔽_{p}) ⊗ H^{p}(W, 𝔽_{p}).

So using that p is odd one gets explicitly:

E_{2}^{∗,∗} ≅ ∧^{∗}(V^{∗}) ⊗ 𝔽_{p}[β(V^{∗})] ⊗ ∧^{∗}(W^{∗}) ⊗ 𝔽_{p}[β(W^{∗})]. (6)

As before, we are using basis free notation, for example V^{∗} = H^{1}(V; 𝔽_{p}) and β is the Bockstein.

The dual of the p-power map ϕ gives us an isomorphism ϕ^{∗} : V^{∗} → W^{∗} so given a basis E = {e_{1}, … e_{n}} for V^{∗}, one can use ϕ^{∗}(E) = {x_{1}, …, x_{n}} as a basis for W.

By standard comparisons it is easy to see that d_{2}|_{W^{∗}} = d_{2}|_{β(W^{∗})} = d_{2}|_{β(V^{∗})} = 0 while

d_{2}(e_{i}) = (the ith component of the extension element) = β(x_{i}) + br^{∗}(x_{i})

for all i. Here we are using the form of the extension element as presented in the end of section 2.1. As ϕ^{∗}(e_{i}) = x_{i}, one has

d_{2}|_{V^{∗}} = β ∘ ϕ^{∗} + br^{∗} ∘ ϕ^{∗}.

Let us calculate E_{3}^{∗,∗}.

Let

A = ∧^{∗}(X_{1}, …, X_{n}) ⊗ 𝔽_{p}[Y_{1}, …, Y_{n}] ⊗ ∧^{∗}(T_{1}, …, T_{n}) ⊗ 𝔽_{p}[S_{1}, …, S_{n}]

be an abstract free graded-commutative algebra where the polynomial generators are degree 2 and the exterior generators are degree 1. Since this is a free object, the assignment X_{i} → x_{i}, T_{i} → e_{i}, S_{i} → β(e_{i}), Y_{i} → β(x_{i}) + br^{∗}(x_{i}) defines a map Ψ of graded-algebras from A to E_{2}^{∗,∗}. As β(x_{i})+br^{∗}(x_{i}) differs form β(x_{i}) by a nilpotent element it is easy to see by induction on the grading that Ψ is an isomorphism. The induced differential D_{2} on A of d_{2} under this isomorphism has D_{2}(X_{i}) = D_{2}(Y_{i}) = D_{2}(S_{i}) = 0 and D_{2}(T_{i}) = Y_{i}. Applying Künneth’s Theorem, we then see that the cohomology of (A, D_{2}) is isomorphic to ∧^{∗}(X_{1}, …, X_{n}) ⊗ 𝔽_{p}[S_{1}, …, S_{n}]. Using this, we see that E_{3}^{∗,∗} is given by

E_{3}^{∗,∗} ≅ ∧^{∗}(W^{∗}) ⊗ 𝔽_{p}[β(V^{∗})] (7)

We have obviously that d_{3}|_{W^{∗}} = 0 and we also have (see for example page 155 of [Be]) :

\begin{aligned}d_{3} ∘ β|_{V^{∗}} & = \{β ∘ d_{2}|_{V^{∗}}\} ∈ E_{3}^{3,0} = ∧^{3}(W^{∗}) \\ & = \{β ∘ (β ∘ ϕ^{∗} + br^{∗} ∘ ϕ^{∗})|_{V^{∗}}\} \\ & = \{β ∘ br^{∗} ∘ ϕ^{∗}|_{V^{∗}}\} since β ∘ β = 0. \end{aligned} (8)

Now note that β is a derivation and that β(x_{i}) is identified with −br^{∗}(x_{i}) in E_{3}^{∗,0} = ∧^{∗}(x_{1}, …, x_{n}) hence β induces the same map as −br^{∗} on E_{3}^{∗,0}. Thus we see that the final expression in equation 8 becomes −br^{∗} ∘ br^{∗} ∘ ϕ^{∗}|_{V^{∗}}. As ϕ^{∗} is an isomorphism, we can conclude that d_{3}|_{β(V^{∗})} = 0 and hence d_{3} = 0 if and only if br^{∗} ∘ br^{∗} = 0 on W^{∗} = ∧^{1}(W^{∗}) or in other words if and only if (W^{∗}, br^{∗}) is a co-Lie algebra. Hence we conclude:

Lemma 2.11. d_{3} = 0 if and only if Log(G) = (W, br) is a Lie algebra.

When this happens E_{3} = E_{4} but then for grading reasons, d_{r}|_{β(V^{∗})} = d_{r}|_{W^{∗}} = 0 for all r ≥ 4. Hence as the differentials are derivations which vanish on the generators, all further differentials in the spectral sequence must be 0. We have proven

Proposition 2.12. E_{3} = E_{∞} if and only if Log(G) is a Lie algebra. In this case,

E_{3}^{∗,∗} = E_{∞}^{∗,∗} = ∧^{∗}(W^{∗}) ⊗ 𝔽_{p}[β(V^{∗})].

Due to the free nature of this graded ring, one can then show in a routine manner that

H^{∗}(G; 𝔽_{p}) = ∧^{∗}(W^{∗}) ⊗ 𝔽_{p}[S^{∗}]

where we have abused notation a bit and identified W^{∗} with its image under the map π^{∗} : H^{∗}(W; 𝔽_{p}) → H^{∗}(G; 𝔽_{p}). S^{∗} is a subspace which restricts isomorphically to the subspace β(V^{∗}) of H^{∗}(V; 𝔽_{p}). Hence we have proved theorem 2.10.

Let us define LGrp to be the full subcategory of BGrp whose objects are the bracket groups G where the associated Log(G) is a Lie algebra. Let Lie be the full subcategory of Brak whose objects are the Lie algebras. Then restricting the natural functors we had before we see LGrp and Lie are naturally equivalent categories.

2.4. A formula for the Bockstein on H^{∗}(G; 𝔽_{p}). From now on we consider groups G ∈ Obj(LGrp). By the previous theorem

H^{∗}(G; 𝔽_{p}) = ∧^{∗}(W^{∗}) ⊗ 𝔽_{p}[S^{∗}]

where W = G/Ω_{1}(G).

We wish to study the Bockstein β : H^{∗}(G; 𝔽_{p}) → H^{∗+1}(G, 𝔽_{p}). The reason for this is that one can determine the groups H^{∗}(G; ℤ) from knowledge of β and the “higher Bocksteins” on H^{∗}(G; 𝔽_{p}).

Recall that β is a derivation, i.e. , for homogeneous elements u, v ∈ H^{∗}(G; 𝔽_{p}) we have

β(uv) = β(u)v + (−1)^{deg(u)}uβ(v).

Also recall that β ∘ β = 0. Since β is a derivation we need only describe it on the generating subspaces W^{∗} and S^{∗} of H^{∗}(G; 𝔽_{p}).

Now recall the central short exact sequence

1 → Ω_{1}(G) = V →^{i} G →^{π} W → 1

Then we have seen that the subalgebra ∧^{∗}(W^{∗}) of H^{∗}(G; 𝔽_{p}) is the image of the map π^{∗} : H^{∗}(W; 𝔽_{p}) → H^{∗}(G; 𝔽_{p}). In the L.H.S.-spectral sequence of the last section, we can identify this image with E_{∞}^{∗,0} = E_{3}^{∗,0}. We saw there that β agrees with −br^{∗} on this subalgebra.

Thus β|_{W^{∗}} = −br^{∗}. It is easy to check that the differential complex (∧^{∗}(W^{∗}), β) is the standard Koszul resolution used to calculate the Lie algebra cohomology H^{∗}(𝔏; 𝔽_{p}), where 𝔏 = (W, br). Also, we see that one can obtain the Lie algebra structure of 𝔏 from knowledge of β on the exterior algebra part of H^{∗}(G; 𝔽_{p}).

Now we are left with finding β|_{S^{∗}} which is harder. Let us develop some more notation. Purely from considering degree restrictions one has:

β|_{S^{∗}} : S^{∗} → (W^{∗} ⊗ S^{∗}) ⊕ ∧^{3}(W^{∗}).

Thus we can write

β|_{S^{∗}} = β_{1} + β_{2} (9)

where β_{1} : S^{∗} → W^{∗} ⊗ S^{∗} and β_{2} : S^{∗} → ∧^{3}(W^{∗}). If we let j = β^{−1} ∘ i^{∗} : S^{∗} → V^{∗} be the natural isomorphism, we can define β̂_{1} = (1⊗j)∘β_{1} ∘j^{−1} : V^{∗} → (W^{∗} ⊗V^{∗}) and β̂_{2} = β_{2} ∘ j^{−1} : V^{∗} → ∧^{3}(W^{∗}).

Before we go on, we should note an inherent ambiguity. Note β̂_{1} and β̂_{2} determine β, however their definition requires a choice for the subspace S^{∗}. We would like that all our expressions involving β be determined from knowledge of the underlying Lie algebra 𝔏 = Log(G). Note W = G/Ω_{1}(G) can be identified with the underlying vector space of 𝔏 and hence so can V = Ω_{1}(G) via the p-power isomorphism ϕ. Thus the subalgebra ∧^{∗}(W^{∗}) of H^{∗}(G; 𝔽_{p}) is unambiguously determined by 𝔏. However the subspace S^{∗} which generates the polynomial part of H^{∗}(G; 𝔽_{p}) is only determined by the fact it restricts isomorphically to the natural subspace β(V^{∗}) of H^{∗}(V; 𝔽_{p}). We can see that given a homomorphism μ : S^{∗} → ∧^{2}(W^{∗}), then the image of S^{∗} under η = Id + μ works just as well, and that all possible other choices for this generating subspace arise in this way. Let us see how changing ones choices effects the decomposition of β in equation 9.

It is easiest to see this by using a basis {s̄_{1}, … s̄_{n}} for the space S^{∗}. If we put this basis in a column vector:

s = (s̄_{1} ; ⋮ ; s̄_{n})

then we have that

β(s) = ξs + β_{2}(s)

where ξ is an n × n matrix with entries in ∧^{1}(W^{∗}).

If we change our choice of subspace S^{∗} using the homomorphism μ mentioned above, we get a new subspace S̄^{∗} = η(S^{∗}) with basis s′ = s + μ(s). We then calculate:

β(s′) = β(s + μ(s)) = ξs + β_{2}(s) + β ∘ μ(s) = ξs′ + (β_{2}(s) + (−br^{∗} − ξ)(μ(s))).

(Here we have used again that β = −br^{∗} on ∧^{∗}(W^{∗}).) So if we decompose β = β′_{1} + β′_{2} using this new subspace S̄^{∗} and note that (1 ∧ μ) ∘ β_{1}(s) = ξμ(s), we see that:

(1 ⊗ η^{−1}) ∘ β′_{1} ∘ η = β_{1} and β′_{2} ∘ η = β_{2} + (−br^{∗} ∘ μ − (1 ∧ μ) ∘ β_{1}) (10)

or

β̂′_{1} = β̂_{1} and β̂′_{2} = β̂_{2} + (−br^{∗} ∘ μ − (1 ∧ μ) ∘ β̂_{1}) (11)

(Here we have made the natural identification of μ as a map μ : V^{∗} → ∧^{2}(W^{∗}).) Thus β̂_{1} is not ambiguous while β̂_{2} depends on the particular subspace S^{∗} chosen, as indicated in equation 11. We will return to this later.

Let us now concentrate on β̂_{1} which is well defined given the Lie algebra 𝔏. Since β ∘ β = 0 we have:

0 = β ∘ β|_{S^{∗}} = β ∘ β_{1} + β ∘ β_{2}. (12)

However since β is a derivation and β = −br^{∗} on ∧^{∗}(W^{∗}) , it follows easily from equation 12 that

0 = (−br^{∗} ⊗ 1) ∘ β_{1} − (1 ⊗ β_{1}) ∘ β_{1} − (1 ⊗ β_{2}) ∘ β_{1} − br^{∗} ∘ β_{2}. (13)

Taking components in the usual way, we get the equations:

(−br^{∗} ⊗ 1) ∘ β_{1} = (1 ⊗ β_{1}) ∘ β_{1} (14)

and

(1 ⊗ β_{2}) ∘ β_{1} = −br^{∗} ∘ β_{2}. (15)

Equation 14 gives us the following commutative diagram:

V^{∗}  ⟶^{−β̂_{1}} W^{∗} ⊗ V^{∗}
−β̂_{1}↓  ↓_{1⊗−β̂_{1}}
W^{∗} ⊗ V^{∗} ⟶^{br^{∗}⊗1} ∧^{2}(W^{∗}) ⊗ V^{∗}

Thus, −β̂_{1} equips V^{∗} with the structure of a comodule over the co-Lie algebra (W^{∗}, br^{∗}). Thus, taking duals, we see that −β̂_{1}^{∗} equips V with the structure of a module over the Lie algebra (W, br). However, we can naturally identify V with W using the p-power map, thus for every Lie algebra 𝔏 = (W, br) we obtain a map λ : 𝔏 → 𝔤𝔩 (𝔏) from the module structure above. Furthermore it follows, that if we give 𝔤𝔩 (𝔏) the canonical Lie algebra structure, then this map λ will be a map of Lie algebras.

These maps λ obtained above are natural in the following sense:

Claim 2.13. Let 𝔏, 𝔏′ ∈ Obj(Lie) and ψ ∈ Mor(𝔏, 𝔏′) then

[λ′(ψ(x))](ψ(y)) = ψ([λ(x)](y)).

(The notation [λ(x)](y) means of course λ(x) ∈ 𝔤𝔩 (𝔏) applied to the element y.)

Proof. Suppose 𝔏 = (W, br) and 𝔏′ = (W̄ , b̄r). Then ψ : W → W̄ induces a unique morphism Exp(ψ) : Exp(𝔏) → Exp(𝔏′) in the category LGrp. This is a class of group homomorphisms where any two representatives of this class induce the same mapping on the Ω_{1} level as mentioned earlier. It is easy to check that any representative of this class will induce the dual map ψ^{∗} from

W̄^{∗} = H^{1}(Exp(𝔏′); 𝔽_{p}) → W^{∗} = H^{1}(Exp(𝔏); 𝔽_{p}).

Since the Bockstein is a natural operation, it will commute with any such map on cohomology and from this, one obtains the following commutative diagram:

V̄^{∗} ⟶^{− ¯β̂_{1}} W̄^{∗} ⊗ V̄^{∗}
ψ^{∗}↓  ↓^{ψ^{∗}⊗ψ^{∗}}
V^{∗} ⟶^{−β̂_{1}} W^{∗} ⊗ V^{∗}

The claim follows easily now by taking the dual of the diagram above and recalling that the map λ was defined via −β̂_{1}^{∗}.

Notice that knowledge of the map λ is equivalent to knowledge of the component β̂_{1} of the Bockstein. We have isolated enough properties of the maps λ and will now set out to determine them explicitly.

2.5. Self-representations. In this section, let k be a finite field with char(k) ≠ 2. Let Lie be the category of finite dimensional Lie algebras over k with morphisms the lie algebra maps. (We will of course be interested in the case k = 𝔽_{p}).

Definition 2.14. A natural self-representation on Lie is a collection of linear maps κ_{𝔏} : 𝔏 → 𝔤𝔩 (𝔏), which satisfy the naturality condition

[κ_{𝔏′}(ψ(x))](ψ(y)) = ψ([κ_{𝔏}(x)](y))

for all 𝔏, 𝔏′ ∈ Obj(Lie), ψ ∈ Mor(𝔏, 𝔏′) and x, y ∈ 𝔏. One says that a selfrepresentation is strong if the maps κ_{𝔏} : 𝔏 → 𝔤𝔩 (𝔏) are maps of Lie algebras.

Examples:

(a) If we set κ = 0 we see easily that we get a strong self-representation which we call the zero representation.

(b) By claim 2.13, we see that the maps λ fit together to give a strong natural self-representation which we will call λ.

(c) If we set κ = ad where ad : 𝔏 → 𝔤𝔩 (𝔏) is given by ad(x)(y) = [x, y] for all x, y ∈ 𝔏, then it is easy to see that each map ad is a map of Lie algebras. Furthermore if 𝔏′ is another Lie algebra and ψ : 𝔏 → 𝔏′ is a morphism of Lie algebras then we have for x, y ∈ 𝔏 :

[ad′(ψ(x))](ψ(y)) = [ψ(x), ψ(y)] = ψ([x, y]) = ψ([ad(x)](y)).

So we see this assignment defines a strong natural self-representation which we call the adjoint representation.

Now we will study self-representations so that we can show λ is the adjoint representation. Note that if κ_{0}, κ_{1} are two self-representations, then if they assign the same map 𝔏 → 𝔤𝔩 (𝔏) for some lie algebra 𝔏, (we will say they agree on 𝔏) then they assign the same map for all lie algebras isomorphic to 𝔏. So we will implicitly use this from now on without mention.

Lemma 2.15. If κ_{0}, κ_{1} are two self-representations, then if they agree on a lie algebra 𝔏, they agree on all Lie subalgebras of 𝔏.

Proof. Follows from the naturality of self-representations.

Recall 𝔤𝔩_{n} is the Lie algebra of n × n matrices and 𝔰𝔩_{n} is the lie algebra of n × n matrices with trace zero. Define 𝔑 to be the 3-dimensional Lie algebra with basis {x, y, z} and bracket given uniquely by [x, y] = z, [x, z] = [y, z] = 0. Note that x, z generate a 2-dimensional abelian subalgebra of 𝔑. Define 𝔖 to be the 2dimensional Lie algebra with basis {x, y} and bracket given uniquely by [x, y] = x. Note as char(k) ≠ 2 this Lie algebra embeds into 𝔰𝔩_{2} by the Lie algebra map Ψ where

Ψ(x) = (\begin{matrix}0 & 0 \\ 1 & 0 \end{matrix})

Ψ(y) = (\begin{matrix}1/2 & 0 \\ 0 & −1/2 \end{matrix}) .

Lemma 2.16. Let κ_{0}, κ_{1} be two self-representations, then if κ_{0} and κ_{1} agree on 𝔰𝔩_{2} and 𝔑 then κ_{0} = κ_{1} as self-representations.

Proof. Assume κ_{1}, κ_{0} are two self-representations satisfying the assumptions above. By the Ado-Iwasawa theorem, all Lie algebras embed as subalgebras of 𝔤𝔩_{n} for some n. So by lemma 2.15, to show κ_{1} = κ_{0} one need only show they agree on 𝔤𝔩_{n} for all n. A basis for 𝔤𝔩_{n} is {δ_{i,j} : i, j = 1, … n} where δ_{i,j} is the matrix with entry one at the (i,j)-position and zero elsewhere. One calculates easily that:

[δ_{i,j}, δ_{l,m}] = \begin{cases}0 & if j ≠ l, m ≠ i \\ δ_{i,m} & if j = l, m ≠ i \\ −δ_{l,j} & if j ≠ l, m = i \\ δ_{i,i} − δ_{j,j} & if j = l, m = i. \end{cases}

Let 𝔏_{ij,lm}^{n} be the Lie subalgebra of 𝔤𝔩_{n} generated by δ_{i,j} and δ_{l,m} for all i, j, l, m ∈ {1, …, n}. Suppose we have shown κ_{1} and κ_{0} agree on 𝔏_{ij,lm}^{n} for all i, j, l, m ∈ {1, …, n}. Then by an easy linearity argument it follows that κ_{1} and κ_{0} agree on 𝔤𝔩_{n} for all n and hence are equal. So we see that if we can show κ_{1} and κ_{0} agree on 𝔏_{ij,lm}^{n} for all n ∈ ℕ, i, j, l, m ∈ {1, …, n} we are done. (Since 𝔤𝔩_{n} ⊂ 𝔤𝔩_{n+1} we see we can assume n ≥ 5 say.) Now let us identify the Lie algebras 𝔏_{ij,lm}^{n}. We have the following cases:

1. 𝔏_{ij,lm} where j ≠ l, m ≠ i. Here we have [δ_{i,j}, δ_{l,m}] = 0 so this Lie algebra is an abelian Lie algebra of dimension ≤ 2. Hence it is contained inside 𝔑.

2. 𝔏_{ij,jm} where m ≠ i, m ≠ j, j ≠ i. This Lie algebra has basis {δ_{i,j}, δ_{j,m}, δ_{i,m}} and has bracket relations, [δ_{i,j}, δ_{j,m}] = δ_{i,m}, [δ_{i,j}, δ_{i,m}] = [δ_{j,m}, δ_{i,m}] = 0. Thus it is easy to see this Lie algebra is isomorphic to 𝔑.

3. 𝔏_{ij,jj} where j ≠ i. Here we have [δ_{i,j}, δ_{j,j}] = δ_{i,j}. So we see this Lie algebra is isomorphic to 𝔖 and so is inside 𝔰𝔩_{2}.

4. 𝔏_{jj,jm} where m ≠ j. We see easily we get the same case as in case 3.

5. 𝔏_{ij,li} where j ≠ l. Here we note that 𝔏_{ij,li} = 𝔏_{li,ij} so it occurs as one of the cases 2 – 4.

6. 𝔏_{ij,ji} where i ≠ j. Here we have that 𝔏_{ij,ji} has basis {δ_{i,j}, δ_{j,i}, δ_{i,i} −δ_{j,j} = Δ}. The bracket is given by [δ_{i,j}, δ_{j,i}] = Δ, [Δ, δ_{i,j}] = δ_{i,j} + δ_{i,j} = 2δ_{i,j} and [Δ, δ_{j,i}] = −δ_{j,i} − δ_{j,i} = −2δ_{j,i} and so we see easily that this Lie algebra is isomorphic to 𝔰𝔩_{2}.

7. 𝔏_{ii,ii}. Obviously this Lie algebra is 1-dimensional and hence lies in either 𝔰𝔩_{2} or 𝔑.

All the Lie algebras 𝔏_{ij,lm}^{n} fit into one of these cases and hence embed in either 𝔰𝔩_{2} or 𝔑. As κ_{1} and κ_{0} agree on these two Lie algebras by assumption, they must agree on all the 𝔏_{ij,lm}^{n} and hence must be equal as argued before.

Now we will study how a self-representation must look in certain particular cases.

Lemma 2.17. Let κ be a self-representation. If x, y ∈ 𝔏 have [x, y] = 0 then [κ(x)](y) = 0. So in particular κ = 0 on abelian Lie algebras.

Proof. By naturality, it is enough to show κ = 0 on abelian Lie algebras of dimension ≥ 2. Fix 𝔏 an abelian Lie algebra of dimension n ≥ 2. Let γ ∈ k − {0, 1}. Then multiplication by γ induces an automorphism Γ of the Lie algebra 𝔏. Given x, y ∈ 𝔏 then if [κ(x)](y) = z it follows from naturality that

\begin{matrix}[κ(Γ(x))](Γ(y)) & = Γ(z) \\ γ^{2}z = γz \end{matrix}

However, γ^{2} ≠ γ so z = [κ(x)](y) = 0. Since x, y were arbitrary, the lemma follows.

Corollary 2.18. Let κ be a self-representation. Then for a Lie algebra 𝔏 and x, y ∈ 𝔏 one has [κ(x)](x) = 0 and [κ(x)](y) = −[κ(y)](x).

Proof. As [x, x] = 0, [κ(x)](x) = 0 follows from lemma 2.17. The second part follows easily from [κ(x + y)](x + y) = 0.

Recall 𝔖 is the Lie algebra with basis {x, y} and [x, y] = x. Let π : 𝔖 → 𝔖 be a linear map given by π(x) = 0 and π(y) = y. It is easy to check that π is a map of Lie algebras.

Lemma 2.19. Let κ be a self-representation. Let 𝔖 be the Lie algebra with basis {x, y} and bracket given by [x, y] = x. Then there is a number a(κ) ∈ k such that one has [κ(u)](v) = a(κ)[u, v] for all u, v ∈ 𝔖.

Proof. By corollary 2.18, one sees it is enough to show [κ(x)](y) = a(κ)x for some a(κ) ∈ k. One knows a priori that

[κ(x)](y) = a(κ)x + b(κ)y, (16)

so we want to show b(κ) = 0. Applying the map of Lie algebras π, mentioned in the paragraph preceding the lemma, to equation 16 and using naturality, we get 0 = b(κ)y and so it follows that b(κ) = 0.

Recall 𝔰𝔩_{2} has basis {δ_{1,2}, δ_{2,1}, δ_{1,1} − δ_{2,2}}. Let us relabel as follows: H = δ_{1,1} − δ_{2,2}, X_{+} = δ_{1,2} and X_{−} = δ_{2,1} then it is easy to verify that the bracket on 𝔰𝔩_{2} is given by

[X_{+}, X_{−}] = H, [H, X_{+}] = 2X_{+} and [H, X_{−}] = −2X_{−}. (17)

Define linear maps Ψ_{mix} : 𝔰𝔩_{2} → 𝔰𝔩_{2}, Ψ_{neg} : 𝔰𝔩_{2} → 𝔰𝔩_{2} as follows:

Ψ_{neg}(H) = H, Ψ_{neg}(X_{+}) = −X_{+}, Ψ_{neg}(X_{−}) = −X_{−}, (18)

and

Ψ_{mix}(H) = H + 2X_{+}, Ψ_{mix}(X_{+}) = −X_{+}, Ψ_{mix}(X_{−}) = H + X_{+} − X_{−}. (19)

It is easy to see that Ψ_{neg} is a map of Lie algebras, and a routine calculation shows Ψ_{mix} is one as well. Now we are ready to prove:

Lemma 2.20. Let κ be a self-representation. Then there is a(κ) ∈ k which is the same as in lemma 2.19, such that κ = a(κ)ad on 𝔰𝔩_{2}, i.e., [κ(x)](y) = a(κ)[x, y] for all x, y ∈ 𝔰𝔩_{2}. Furthermore, if κ is strong, then a(κ) = 0 or 1.

Proof. One has [κ(H)](H) = [κ(X_{+})](X_{+}) = [κ(X_{−})](X_{−}) = 0 as usual. Now H, X_{+} span a Lie subalgebra of 𝔰𝔩_{2} which is isomorphic to 𝔖 via H → −2y and X_{+} → x. Thus by naturality and lemma 2.19 we have [κ(H)](X_{+}) = a(κ)[H, X_{+}]. A similar argument works for X_{−} in place for X_{+}. So we conclude easily that

[κ(H)](·) = a(κ)[H, ·]. (20)

Thus it follows that [κ(X_{±})](H) = a(κ)[X_{±}, H]. Now [κ(X_{+})](X_{−}) = αH +βX_{+} + γX_{−}. Applying the map of Lie algebras Ψ_{neg} to this equation, and using naturality one gets:

\begin{matrix}[κ(Ψ_{neg}(X_{+}))](Ψ_{neg}(X_{−})) = & αH − βX_{+} − γX_{−} \\ αH + βX_{+} + γX_{−} = & αH − βX_{+} − γX_{−}. \end{matrix}

So β = γ = 0. Thus [κ(X_{+})](X_{−}) = αH = α[X_{+}, X_{−}]. So if we can show α = a(κ) we can conclude that

[κ(X_{±})](·) = a(κ)[X_{±}, ·]

and together with equation 20 this implies κ = a(κ)ad on 𝔰𝔩_{2}. So it remains only to show α = a(κ). Applying the Lie algebra map Ψ_{mix} to the equation [κ(X_{+})](X_{−}) = αH one gets:

\begin{matrix}[κ(Ψ_{mix}(X_{+}))](Ψ_{mix}(X_{−})) = & αΨ_{mix}(H) \\ −[κ(X_{+})](H) + [κ(X_{+})](X_{−}) = & α(H + 2X_{+}) \\ −a(κ)[X_{+}, H] + αH = & αH + 2αX_{+} \\ 2a(κ)X_{+} = & 2αX_{+}. \end{matrix}

From which one concludes that a(κ) = α as desired.

Now if κ = a(κ)ad is a map of Lie algebras, then we have

\begin{aligned}κ([X_{+}, X_{−}]) & = κ(X_{+}) ∘ κ(X_{−}) − κ(X_{−}) ∘ κ(X_{+}) \\ a(κ)ad([X_{+}, X_{−}]) & = a(κ)^{2}ad(X_{+}) ∘ ad(X_{−}) − a(κ)^{2}ad(X_{−}) ∘ ad(X_{+}) \\ a(κ)ad([X_{+}, X_{−}]) & = a(κ)^{2}ad([X_{+}, X_{−}]). \end{aligned}

Since ad([X_{+}, X_{−}]) is nonzero, it follows that a(κ) = a(κ)^{2} or in other words, that a(κ) = 0 or 1.

Recall 𝔑 has basis {x, y, z} and bracket given by [x, z] = [y, z] = 0, [x, y] = z. Define linear maps Ψ_{x}, Ψ_{y} : 𝔑 → 𝔑 by

Ψ_{x}(y) = Ψ_{x}(z) = 0, Ψ_{x}(x) = x

and

Ψ_{y}(x) = Ψ_{y}(z) = 0, Ψ_{y}(y) = y.

It is easy to verify that these maps are maps of Lie algebras.

Lemma 2.21. Let κ be a self-representation. Then there is μ(κ) ∈ k such that κ = μ(κ)ad on 𝔑.

Proof. As z is central one sees by lemma 2.17, that κ(z) = 0 = ad(z). Now [κ(x)](y) = αx + βy + μz. Applying the Lie algebra map Ψ_{x} to this equation and using naturality one gets 0 = αx. So one concludes α = 0. Similarly using Ψ_{y} instead one concludes β = 0. So [κ(x)](y) = μz = μ[x, y]. So now it is easy to see [κ(x)](·) = μ[x, ·] and [κ(y)](·) = μ[y, ·] via lemma 2.17 and corollary 2.18. Thus setting μ(κ) = μ we are done.

Recall 𝔰𝔬_{n} ⊂ 𝔤𝔩_{n}, the subset of skew-symmetric n × n matrices, is a Lie subalgebra. For char(k) ≠ 2, 𝔰𝔬_{3} is 3-dimensional with basis {X = δ_{1,2}−δ_{2,1}, Y = δ_{1,3}−δ_{3,1}, Z = δ_{2,3} − δ_{3,2}} and bracket given by the relations [X, Y] = −Z,[Y, Z] = −X,[Z, X] = −Y. As we are over a finite field of odd characteristic, 𝔰𝔬_{3} is isomorphic to 𝔰𝔩_{2} via the map which sends X → \frac{X_{+}−X_{−}}{2} , Y → a\frac{X_{+}+X_{−}}{2} + b\frac{H}{2} , Z → b\frac{X_{+}+X_{−}}{2} − a\frac{H}{2} . Here a, b ∈ k are such that a^{2} + b^{2} = −1.

Lemma 2.22. Let κ be a self-representation. Then κ = μ(κ)ad = a(κ)ad on 𝔰𝔬_{3} where μ(κ) is the same as that of lemma 2.21 and a(κ) is the same as lemma 2.20. In particular, a(κ) = μ(κ).

Proof. The fact that κ = a(κ)ad follows from lemma 2.20 and the fact that 𝔰𝔬_{3} is isomorphic to 𝔰𝔩_{2}. So it remains to show that κ = μ(κ)ad also. Consider the Lie algebra 𝔤𝔩_{3}. In the proof of lemma 2.16 we saw that

{δ_{i,j}, δ_{j,k}, δ_{i,k}}

formed a basis of a Lie subalgebra isomorphic to 𝔑 for i, j, k distinct. Thus

[κ(δ_{i,j})](δ_{j,k}) = μ(κ)[δ_{i,j}, δ_{j,k}] = μ(κ)δ_{i,k}

for all i, j, k distinct. Also [κ(δ_{i,j})](δ_{l,m}) = 0 if j ≠ l, i ≠ m as under these conditions [δ_{i,j}, δ_{l,m}] = 0. So

[κ(δ_{1,2} − δ_{2,1})](δ_{1,3} − δ_{3,1}) =[κ(δ_{1,2})](δ_{1,3}) − [κ(δ_{1,2})](δ_{3,1})
− [κ(δ_{2,1})](δ_{1,3}) + [κ(δ_{2,1})](δ_{3,1})
=0 − (−μ(κ)δ_{3,2}) − (μ(κ)δ_{2,3}) + 0
=μ(κ)(δ_{3,2} − δ_{2,3})
=μ(κ)[δ_{1,2} − δ_{2,1}, δ_{1,3} − δ_{3,1}],

and similarly

[κ(δ_{1,2} − δ_{2,1})](δ_{2,3} − δ_{3,2}) = μ(κ)[δ_{1,2} − δ_{2,1}, δ_{2,3} − δ_{3,2}]

and

[κ(δ_{1,3} − δ_{3,1})](δ_{2,3} − δ_{3,2}) = μ(κ)[δ_{1,3} − δ_{3,1}, δ_{2,3} − δ_{3,2}].

So with these 3 equations and corollary 2.18 one concludes that κ = μ(κ)ad on 𝔰𝔬_{3}.

We now state:

Theorem 2.23. Let κ be a strong self-representation. If κ ≠ 0 on 𝔰𝔩_{2} then κ = ad as self-representations. If κ = 0 on 𝔰𝔩_{2} then κ = 0 as self-representations.

Proof. If κ ≠ 0 on 𝔰𝔩_{2} then we have a(κ) = μ(κ) = 1 from lemmas 2.20 and 2.22. Thus κ = ad on 𝔰𝔩_{2} and on 𝔑. Thus by lemma 2.16 we have that κ = ad as self-representations. the case κ = 0 on 𝔰𝔩_{2} proceeds similarly.

For use later on, we need to state theorem 2.23 in slightly more generality. To do this we need to introduce the categories Lie_{k}(p) for k ≥ 1 and p a prime. The objects of this category will be Lie algebras over ℤ/p^{k}ℤ, that is free ℤ/p^{k}ℤ-modules R of finite rank equipped with a bilinear, alternating form [·, ·] : R × R → R which satisfies the Jacobi identity. The morphisms will be the ℤ/p^{k}ℤ-module maps which preserve the brackets. Thus Lie_{1}(p) is just the same category Lie considered before for k = 𝔽_{p}.

For t ≤ k, we have reduction functors Lie_{k}(p) → Lie_{t}(p). These reduction functors are obtained in the obvious way from the reduction map of ℤ/p^{k}ℤ to ℤ/p^{t}ℤ. For 𝔏 ∈ Obj(Lie_{k}(p)) we denote 𝔏̄ to be its reduction in Obj(Lie_{1}(p)). We will also use the bar notation for the reduction of a morphism. Note that there is the canonical reduction homomorphism which is a map of abelian groups which preserves the bracket from 𝔏 to 𝔏̄, we will refer to maps of abelian groups which preserves brackets as Lie algebra maps for the rest of this section.

Definition 2.24. A natural self-representation on Lie_{k}(p) is a collection of maps κ_{𝔏} : 𝔏 → 𝔤𝔩(𝔏̄) of abelian groups, which satisfy the naturality condition

[κ_{𝔏′}(ψ(x))](ψ̄(y)) = ψ̄([κ_{𝔏}(x)](y))

for all 𝔏, 𝔏′ ∈ Obj(Lie_{k}(p)), ψ ∈ Mor(𝔏, 𝔏′), x ∈ 𝔏 and y ∈ 𝔏̄. We say the natural self-representation is strong if the maps κ_{𝔏} : 𝔏 → 𝔤𝔩(𝔏̄) are maps of Lie algebras.

Note this definition extends the previous one.

It is easy to check that if we assign κ_{𝔏} : 𝔏 → 𝔤𝔩(𝔏̄) to be the composition of the reduction homomorphism with ad : 𝔏̄ → 𝔤𝔩(𝔏̄) then we obtain a strong selfrepresentation which we will call ad.

The Lie algebras 𝔤𝔩_{n}, 𝔰𝔩_{2},𝔑, 𝔖 and 𝔰𝔬_{3} will denote the obvious Lie algebras in Obj(Lie_{k}(p)). For example, 𝔤𝔩_{n} is the Lie algebra of n × n matrices with ℤ/p^{k}ℤ entries. Theorem 2.23 is true for these more general self-representations. We state this in the next theorem.

Theorem 2.25. Let p be an odd prime and let κ be a strong self-representation over Lie_{k}(p). If κ ≠ 0 on 𝔰𝔬_{3} and 𝔰𝔩_{2}, then κ = ad as self-representations.

Proof. The proof proceeds as before with only some modifications so we leave it to the reader (see [Pak]). The only thing to note is there is a generalized Ado-Iwasawa theorem (see [W4]) which says that any 𝔏 ∈ Obj(Lie_{k}(p)) embeds as a subalgebra of 𝔤𝔩_{n}(ℤ/p^{k}ℤ).

2.6. Bockstein spectral sequence implies λ = ad. Recall we had λ a strong self-representation over 𝔽_{p} which was defined using the Bockstein. In this section we show λ = ad as self-representations.

By Theorem 2.23, it is enough to show λ is not identically zero on 𝔰𝔩_{2}. To do this we will need the Bockstein spectral-sequence.

Let X be a topological space (with finitely generated integral homology). Then the coefficient sequence

0 → ℤ →^{p} ℤ → ℤ/pℤ → 0

gives a long exact sequence

⋯ ⟶ H^{∗}(X; ℤ) ⟶^{p} H^{∗}(X; ℤ) ⟶^{mod} H^{∗}(X; 𝔽_{p}) ⟶^{δ} H^{∗+1}(X, ℤ) ⟶ ⋯

which gives a spectral sequence in a standard way if we think of it as an exact couple. This spectral sequence has B_{1}^{∗} = H^{∗}(X; 𝔽_{p}) with differential given by the Bockstein β (as defined previously) and converges to 𝔽_{p}⊗(H^{∗}(X; ℤ)/torsion). If we apply this to the case X=BG where G is a finite group, then the Bockstein spectral sequence must converge to zero in positive gradings, as the integral cohomology of a finite group is torsion in positive gradings.

Now we will apply the Bockstein spectral sequence to show that λ is not zero on 𝔰𝔩_{2}.

Let {H, X_{+}, X_{−}} be the usual basis for 𝔰𝔩_{2} and let G = Exp(𝔰𝔩_{2}). Let {h, x_{+}, x_{−}} be the obvious dual basis for 𝔰𝔩_{2}. Then as we have seen before,

B_{1}^{∗} = H^{∗}(G; 𝔽_{p}) = ∧^{∗}(h, x_{+}, x_{−}) ⊗ 𝔽_{p}[s_{h}, s_{+}, s_{−}]

where we have chosen the s-basis to correspond in the obvious way. Then we see using the formulas found before that on the exterior algebra part the Bockstein is given by

\begin{aligned}β(h) & = −[·, ·]_{H} = −x_{+}x_{−} \\ β(x_{+}) & = −[·, ·]_{X_{+}} = −2hx_{+} \\ β(x_{−}) & = −[·, ·]_{X_{−}} = 2hx_{−}. \end{aligned}

So as B_{1}^{1} = ∧^{1}(h, x_{+}, x_{−}) one sees that B_{2}^{1} = 0. Now note B_{1}^{2} = ∧^{2}(h, x_{+}, x_{−}) ⊕ span(s_{h}, s_{+}, s_{−}) and by the above formulas, the exterior part contributes nothing to B_{2}^{2}. If λ is zero on 𝔰𝔩_{2} then this means β maps span(s_{h}, s_{+}, s_{−}) into ∧^{3}(h, x_{+}, x_{−}) which is 1-dimensional generated by hx_{+}x_{−}. So we see β must have at least a 2-dimensional kernel on the span of {s_{h}, s_{+}, s_{−}}. Thus dim(B_{2}^{2}) ≥ 2. Now B_{1}^{3} is equal to

∧^{3}(h, x_{+}, x_{−}) ⊕ span({s_{h}h, s_{h}x_{+}, s_{h}x_{−}, s_{+}h, s_{+}x_{+}, s_{+}x_{−}, s_{−}h, s_{−}x_{+}, s_{−}x_{−}}).

We will show β is injective on the second part so that dim(B_{2}^{3}) ≤ 1. Note that β(s_{∗}) = a_{∗}hx_{+}x_{−}, for some a_{∗} ∈ 𝔽_{p}, where * stands for an arbitrary subscript. So β(s_{∗})x_{∗} = 0 and one has:

\begin{matrix}β(s_{∗}x_{∗}) = & β(s_{∗})x_{∗} + s_{∗}β(x_{∗}) \\ = & s_{∗}β(x_{∗}) \end{matrix}

So because {β(x_{h}), β(x_{+}), β(x_{−})} are linearly independent, and {s_{h}, s_{+}, s_{−}} are algebraically independent, it is easy to see β is injective on the second part of B_{1}^{3} as claimed. Thus we have B_{2}^{1} = 0, dim(B_{2}^{2}) ≥ 2 and dim(B_{2}^{3}) ≤ 1. However the Bockstein spectral sequence converges to zero in positive gradings implying we need dim(B_{2}^{2}) ≤ dim(B_{2}^{3}) which we don’t have. Thus we have a contradiction and our assumption that λ = 0 must be false.

Thus we finally conclude λ = ad for all Lie algebras. Now let us chase through the definitions to get formulas for the β_{1} part of β. So fix a Lie algebra 𝔏 over 𝔽_{p}. Fix a basis E = {e_{1}, …, e_{n}} of 𝔏 and let G = Exp(𝔏). Then H^{∗}(G; 𝔽_{p}) = ∧^{∗}(x_{1}, …, x_{n})⊗𝔽_{p}[s_{1}, …, s_{n}] where x_{1}, …, x_{n} is the dual basis of E and s_{1}, …, s_{n} is the corresponding basis for the polynomial part of the cohomology. Let c_{ij}^{k} = [e_{i}, e_{j}]_{k} be the structure constants of 𝔏 with respect to the basis E. We have shown that

[λ(e_{i})](e_{j}) = ∑_{k=1}^{n} c_{ij}^{k}e_{k}.

Recalling that λ is the dual of −β_{1} : S^{∗} → W^{∗} ⊗ S^{∗}, we see easily that

−β_{1}(s_{k}) = ∑_{i,j=1}^{n} c_{ij}^{k}x_{i}s_{j}.

Using that c_{ij}^{k} = −c_{ji}^{k}, one can easily deduce the following equation for β:

β(s_{k}) = ∑_{i,j=1}^{n} c_{ij}^{k}s_{i}x_{j} + β_{2}(s_{k}). (21)

Thus we can also recover the Lie algebra structure of 𝔏 from knowledge of the β_{1} component of β|_{S^{∗}}. It remains to study β_{2} which is the remaining term of the Bockstein. Before doing that, we will digress to give an application of equation 21.

2.7. Comodule algebra structure. Let G ∈ Obj(LGrp) then the multiplication in G induces a homomorphism μ : Ω_{1}(G) × G → G and

Δ = μ^{∗} : H^{∗}(G; 𝔽_{p}) → H^{∗}(Ω_{1}(G); 𝔽_{p}) ⊗ H^{∗}(G; 𝔽_{p})

gives H^{∗}(G; 𝔽_{p}) the structure of a comodule algebra over the Hopf algebra H^{∗}(Ω_{1}(G); 𝔽_{p}). (See [W2].)

In order to describe Δ, it is enough to describe it on algebra generators of H^{∗}(G; 𝔽_{p}). To do this let us write

H^{∗}(G; 𝔽_{p}) = ∧^{∗}(x_{1}, …, x_{n}) ⊗ 𝔽_{p}[s_{1}, …, s_{n}]

and H^{∗}(Ω_{1}(G); 𝔽_{p}) = ∧^{∗}(t_{1}, …, t_{n}) ⊗ 𝔽_{p}[s̄_{1}, …, s̄_{n}] with s̄_{i} = β(t_{i}) for all i. Furthermore we have made our choices so that j^{∗}(s_{i}) = s̄_{i} where j is the inclusion map of Ω_{1}(G) into G and also so that equation 21 holds. Recall that the Hopf algebra structure of Ω_{1}(G) is one where all the generators s̄_{i} and t_{i} are primitive. Now we can state:

Corollary 2.26. Let G ∈ Obj(LGrp) and c_{ij}^{k} be the structure constants of Log(G) in a suitable basis. Using the notation of the paragraph above, one has the following formulas which determine Δ:

Δ(x_{k}) =1 ⊗ x_{k}
Δ(s_{k}) =s̄_{k} ⊗ 1 + 1 ⊗ s_{k} + ∑_{i,j} c_{ij}^{k}t_{i} ⊗ x_{j}

Proof. Precomposing μ with the natural inclusions of Ω_{1}(G) and G into Ω_{1}(G)×G, it is not hard to show that Δ = μ^{∗} must take the following form on the generators:

Δ(x_{k}) =1 ⊗ x_{k}
Δ(s_{k}) =s̄_{k} ⊗ 1 + 1 ⊗ s_{k} + ∑_{i,j} a_{ij}^{k}t_{i} ⊗ x_{j}.

Thus it is enough to show a_{ij}^{k} = c_{ij}^{k} for all i, j, k. As the Bockstein β is a natural operation, and since Δ is induced from a homomorphism of groups, they must commute, that is Δ ∘ β = (β ⊗ 1 + 1 ⊗ β) ∘ Δ. Applying β ⊗ 1 + 1 ⊗ β to the formula for Δ(s_{k}) and using equation 21, we get

(β ⊗ 1 + 1 ⊗ β)(Δ(s_{k})) = 1 ⊗ β(s_{k}) + ∑_{i,j=1}^{n} a_{ij}^{k}(s̄_{i} ⊗ x_{j} − t_{i} ⊗ β(x_{j})) (22)

but

Δ(β(s_{k})) = ∑_{l,m=1}^{n} c_{lm}^{k}Δ(s_{l}x_{m}) + Δ(β_{2}(s_{i}))

= ∑_{l,m=1}^{n} c_{lm}^{k}(s̄_{l} ⊗ x_{m} + 1 ⊗ s_{l}x_{m} + ∑_{u,v} a_{uv}^{l}t_{u} ⊗ x_{v}x_{m}) + 1 ⊗ β_{2}(s_{i})

Now we know the expressions in equation 22 and the last one are equal so equating their H^{2}(Ω_{1}(G); 𝔽_{p}) ⊗ H^{1}(G; 𝔽_{p}) components we get:

∑_{i,j=1}^{n} a_{ij}^{k}s̄_{i} ⊗ x_{j} = ∑_{l,m=1}^{n} c_{lm}^{k}s̄_{l} ⊗ x_{m}

which immediately gives a_{ij}^{k} = c_{ij}^{k} for all i, j, k which is what we desired.

Thus we see that the comodule algebra structure of H^{∗}(G; 𝔽_{p}) also determines the Lie algebra corresponding to G.

2.8. The class [η] and lifting uniform towers. In this section we will show that the β_{2} component for the Bockstein, defines a cohomology class in a suitable cohomology group.

Fix 𝔏 = (W, br) ∈ Obj(Lie). Recall the existence of a Koszul complex:

0 → ∧^{0}(𝔏, ad) →^{d} … →^{d} ∧^{n}(𝔏, ad) → 0

whose cohomology is H^{∗}(𝔏; ad). Here ∧^{i}(𝔏, ad) denotes the 𝔏-valued alternating i-forms on 𝔏 and d is given by

\begin{aligned}(dω)(u_{0}, …, u_{l}) = & ∑_{i<j}(−1)^{i+j}ω([u_{i}, u_{j}], u_{0}, …, û_{i}, …, û_{j}, …, u_{l}) \\ & + ∑_{i=0}^{l}(−1)^{i}[u_{i}, ω(u_{0}, …, û_{i}, …, u_{l})] \end{aligned}

on l-forms ω. If we view ω : ∧^{l}(W) → W and set ad : W ⊗ W → W to be the adjoint action map, one can show easily that

dω = −ω ∘ br + ad ∘ (1 ∧ ω).

Using β̂_{2} instead of β_{2} in equation 15 and taking duals one obtains:

0 = −(−β̂_{2}^{∗}) ∘ br + (−β̂_{1}^{∗}) ∘ (1 ∧ −β̂_{2}^{∗}).

However, we have shown that −β̂_{1}^{∗} = ad and so we can conclude that −β̂_{2}^{∗} : ∧^{3}(W) → V is a closed 3-form in the Koszul resolution for H^{∗}(𝔏; ad). In a similar fashion, when one looks at the effect of changing the choice of the subspace S^{∗} via a map μ : V^{∗} → ∧^{2}(W^{∗}) as in equation 11, one sees that the net result is to add d(−μ^{∗}) to −β̂_{2}^{∗}. So although the 3-form −β̂_{2}^{∗} is not uniquely determined from 𝔏, it is uniquely determined up to the addition of a boundary in the Koszul resolution above and hence defines a unique cohomology class [η] ∈ H^{3}(𝔏; ad). In fact it is easy to see that one can get any representative of this cohomology class by suitable choice of μ.

So now it remains to study [η] ∈ H^{3}(𝔏; ad). It is true for p ≠ 3 that [η] = 0 if and only if 𝔏 lifts to a ℤ/p^{2}ℤ Lie algebra. This requires a bit of work using results of lifting Lie algebras as mentioned in [Pak] and T. Weigel’s Exp-Log correspondence. We will first show that [η] = 0 if and only if G = Exp(𝔏) has an Ω_{1}-extension with the ΩEP (these terms will be defined later). This on the other hand is quite elementary and does not need knowledge of the Exp-Log correspondence.

We now discuss various concepts necessary to see this fact. First recall definitions 1.1 and 1.2. Furthermore we define:

Definition 2.27. G is powerful if [G, G] ⊆ G^{p}.

Definition 2.28. A tower of groups will be a sequence of p-central groups

{G_{1}, …, G_{N}}

(where N = 1, 2, …, ∞) equipped with surjective homomorphisms π_{i} : G_{i} → G_{i−1} with kernel Ω_{1}(G_{i}) for i = 1, …, N. (G_{0} = 1 by convention.)

Note the definition implies G_{1} = Ω_{1}(G_{1}) and hence is elementary abelian. Also note if we have a tower, we have central short exact sequences of groups

1 → Ω_{1}(G_{i}) → G_{i} →^{π_{i}} G_{i−1} → 1

and these pull back to central short exact sequences

1 → Ω_{1}(G_{i}) → E_{i} →^{π_{i}} Ω_{1}(G_{i−1}) → 1

as considered in the beginning of this paper. Thus these have a p-power map ϕ where ϕ(x) = x̂^{p} for x ∈ Ω_{1}(G_{i−1}). Note the ϕ map is automatically injective by definition of Ω_{1}(G_{i}).

Definition 2.29. A uniform tower of groups is a tower of groups where all the p-power maps ϕ of the tower are isomorphisms.

Thus a uniform tower is where the p-power maps ϕ are also surjective. Note it is well known that if G is p-central then G/Ω_{1}(G) is also p-central so all p-central groups fit into a tower. By induction, it is also easy to see that G_{n} a group in the nth stage of a uniform tower has exponent p^{n} and is powerful.

In general if you have a tower, then G_{2} fits in a short exact sequence

1 → Ω_{1}(G_{2}) → G_{2} → G_{1} → 1

and as G_{1} is elementary abelian, one sees immediately that G_{2} groups in uniform towers are exactly the groups associated to bracket algebras considered before. Now given a tower of length N with end group G_{N} one can ask when it extends to a tower of length N + 1, i.e., when does there exist a p-central group G_{N+1} such that G_{N+1}/Ω_{1}(G_{N+1}) is isomorphic to G_{N}. We will be primarily interested in uniform towers and extending to uniform towers. This question has been considered independently in [W1]. In order to coincide with the notation there we introduce the concept of ΩEP used in [W1] and [W2]. We will provide theorems which were proved independently by us for the uniform case and refer the reader to [W1] and [W2] for lifting in the more general situation.

Definition 2.30. A p-central group G is said to have the ΩEP if there exists a p-central group Ĝ such that Ĝ/Ω_{1}(Ĝ) = G.

Note that if we have such an extension with Ĝ →^{π} G then π^{−1}(Ω_{1}(G)) ⊂ Ĝ is an extension of Ω_{1}(G) which has ϕ map injective. We call such an extension an Ω_{1} extension. Let n = dim(Ω_{1}(G)) and l = dim(Ω_{1}(Ĝ)). In

H^{2}(Ω_{1}(G); Ω_{1}(Ĝ)) = [∧^{∗}(x_{1}, …, x_{n}) ⊗ 𝔽_{p}[s_{1}, …, s_{n}]]^{l}

this extension for π^{−1}(Ω_{1}(G)) is represented by an element of the form

(s_{1} + μ_{1}, …, s_{n} + μ_{n}, μ_{n+1}, …, μ_{l})

with μ_{i} ∈ ∧^{∗}(x_{1}, …, x_{n}). Note that s_{1} + μ_{1}, …, s_{n} + μ_{n} are in the image of the restriction i^{∗} : H^{∗}(G; 𝔽_{p}) → H^{∗}(Ω_{1}(G); 𝔽_{p}). Thus one has n algebraically independent elements in the second grading in the image of the restriction i^{∗}. On the other hand if we had a p-central group such that dim(Ω_{1}(G)) = n and there are n algebraically independent elements of degree two in image i^{∗} : H^{∗}(G; 𝔽_{p}) → H^{∗}(Ω_{1}(G); 𝔽_{p}). Then by choosing the right basis for

H^{2}(Ω_{1}(G); 𝔽_{p}) = ∧^{∗}(x_{1}, …, x_{n}) ⊗ 𝔽_{p}[s_{1}, …, s_{n}]

one has these elements of the form s_{1} + μ_{1}, …, s_{n} + μ_{n} with μ_{i} ∈ ∧^{2}(x_{1}, …, x_{n}). Thus taking an element in H^{2}(G; Ω_{1}(G)) which restricts to (s_{1} + μ_{1}, …, s_{n} + μ_{n}) ∈ H^{2}(Ω_{1}(G); Ω_{1}(G)), we see easily that the group this element represents say Ĝ has Ĝ/Ω_{1}(Ĝ) = G. Thus G has the ΩEP. So we conclude

Lemma 2.31. Let G be a p-central group with n = dim(Ω_{1}(G)). Then G has the ΩEP if and only if there are n algebraically independent elements of degree 2 in image i^{∗} : H^{∗}(G; 𝔽_{p}) → H^{∗}(Ω_{1}(G); 𝔽_{p}).

Now note in a tower {G_{1}, …, G_{N}}, G_{i} has the ΩEP for 1 ≤ i < N by definition. This lets us prove:

Theorem 2.32. Let {G_{1}, …, G_{N}} be a uniform tower with dim(G_{1}) = n. Then for 1 ≤ i < N we have

H^{∗}(G_{i}; 𝔽_{p}) = ∧^{∗}(x_{1}, …, x_{n}) ⊗ 𝔽_{p}[s_{1}, …, s_{n}]

where deg(x_{i}) = 1 (and for i > 1, x_{i} is induced from G_{i−1}) and deg(s_{i}) = 2 (and for i > 1, s_{i} “comes” from Ω_{1}(G_{i})), and H^{∗}(G_{N}; 𝔽_{p}) is given by the same formula if and only if the tower extends to a uniform tower {G_{1}, …, G_{N}, G_{N+1}}.

Proof. By the comments in the preceding paragraph and lemma 2.31 one obtains the first part of this theorem by induction on i where 1 ≤ i < N. More explicitly note that one has the result for G_{1}. So fix i > 1 and assume we have shown the first part for all j < i. Then we have the central short exact sequence:

1 → Ω_{1}(G_{i}) → G_{i} →^{π_{i}} G_{i−1} → 1.

So the E_{2}^{∗,∗} term of the L.H.S.-spectral sequence of this extension is given by

∧^{∗}(t_{1}, …, t_{n}) ⊗ 𝔽_{p}[s_{1}, …, s_{n}] ⊗ ∧^{∗}(x_{1}, …, x_{n}) ⊗ 𝔽_{p}[y_{1}, …, y_{n}]

where the first two terms are the cohomology of Ω_{1}(G_{i}) and the last two terms are the cohomology of G_{i−1} by induction. One has as usual d_{2}(s_{i}) = d_{2}(x_{i}) = d_{2}(y_{i}) = 0 and d_{2}(t_{i}) are the elements representing the extension. By the proof of lemma 2.31 and the fact G_{i} is an Ω_{1} extension of G_{i−1} one sees (after changing basis for y’s) that these elements are of the form d_{2}(t_{i}) = y_{i} + μ_{i} for all i where μ_{i} ∈ ∧^{∗}(x_{1}, …, x_{n}). Thus as in the proof of theorem 2.10, one gets E_{3}^{∗,∗} = 𝔽_{p}[s_{1}, …, s_{n}]⊗∧^{∗}(x_{1}, …, x_{n}) with d_{3}(x_{i}) = 0. Now E_{∞}^{0,∗} is the image of i^{∗} : H^{∗}(G_{i}; 𝔽_{p}) → H^{∗}(Ω_{1}(G_{i}); 𝔽_{p}) and by lemma 2.31 this contains n algebraically independent elements in degree 2. Since E_{2}^{0,2} has dim n we see by necessity that d_{3} = 0 on E_{2}^{0,2}, which implies d_{3} = 0. Then as in theorem 2.10 one sees that H^{∗}(G_{i}; 𝔽_{p}) is as stated. So we are done with the first part. If the tower extends to a uniform tower of length one more, then one can perform the same argument on G_{N} to conclude its cohomology is as stated. On the other hand if its cohomology is as stated, image i^{∗} : H^{∗}(G_{N}; 𝔽_{p}) → H^{∗}(Ω_{1}(G_{N}); 𝔽_{p}) has n algebraically independent elements in degree 2 and so G_{N} has the ΩEP. Using these elements as in the proof of lemma 2.31, one constructs an extension to a uniform tower of length N + 1.

Examples:

Definition 2.33. Fix p an odd prime number, then Γ_{n,k}(p) is defined as the kernel of the reduction homomorphism

GL_{n}(ℤ/p^{k+1}ℤ) → GL_{n}(ℤ/pℤ)

for all n, k ≥ 1. Similarly, Γ̂_{n,k}(p) is the group obtained by replacing GL_{n} with SL_{n} in the definition above.

One shows by induction on k that the Γ_{n,k}(p) are p-groups and it is then easy to show that for fixed n they fit together to give an infinite uniform tower:

→ Γ_{n,k}(p) → Γ_{n,k−1}(p) → ⋯ → Γ_{n,1}(p) → 1.

A similar statement holds for the Γ̂_{n,k}(p).

Theorem 2.32 has the following immediate corollary:

Corollary 2.34. For n, k ≥ 1 and p an odd prime we have:

H^{∗}(Γ_{n,k}(p); 𝔽_{p}) ≅ ∧^{∗}(x_{1}, …, x_{n^{2}}) ⊗ 𝔽_{p}[s_{1}, …, s_{n^{2}}]

and

H^{∗}(Γ̂_{n,k}(p); 𝔽_{p}) ≅ ∧^{∗}(x_{1}, …, x_{n^{2}−1}) ⊗ 𝔽_{p}[s_{1}, …, s_{n^{2}−1}]

where deg(x_{i}) = 1 and deg(s_{i}) = 2 for all i.

Recall that if G ∈ Obj(BGrp) then G = G_{2} of a uniform tower {G_{1}, G_{2}}. This tower extends to a uniform tower of length 3 if and only if Log(G) is a Lie algebra. This follows form theorem 2.32 and theorem 2.10. Next, we will show that this uniform tower {G_{1}, G_{2}} extends to a tower of length 4 if and only if [η] = 0 where [η] ∈ H^{3}(𝔏; ad) is the cohomology class defined earlier.

Now suppose we have a uniform tower {G_{1}, G_{2}, G′_{3}} so G_{2} ∈ Obj(LGrp) and has 𝔽_{p}-cohomology given by theorem 2.10. We will now show that there is a uniform tower {G_{1}, G_{2}, G_{3}, G_{4}} if and only if [η] = 0 where we say G_{3} instead of G′_{3} since we might have to choose a different Ω_{1} extension of G_{2} to do this. Note the tower {G_{1}, G_{2}} extends to one of length 4 as desired if and only if G_{2} has a uniform Ω_{1} extension G_{3} which itself has the ΩEP. By theorem 2.32, this happens if and only if G_{3} has 𝔽_{p}-cohomology ∧^{∗}(x_{1}, …, x_{n}) ⊗ 𝔽_{p}[s_{1}, …, s_{n}] where deg(x_{i}) = 1, deg(s_{i}) = 2 and n = dim(G_{1}).

So let us study the cohomology of an Ω_{1} extension G_{3} of G_{2}. So H^{∗}(G_{2}; 𝔽_{p}) = ∧^{∗}(x_{1}, …, x_{n}) ⊗ 𝔽_{p}[s_{1}, …, s_{n}] where deg(s_{i}) = 2, deg(x_{i}) = 1 as usual. Then G_{3} is represented in H^{2}(G_{2}; Ω_{1}(G_{3})) = [H^{2}(G_{2}; 𝔽_{p})]^{n} by (s_{1}+μ_{1}, …, s_{n}+μ_{n}) where μ_{i} ∈ ∧^{2}(x_{1}, …, x_{n}) under a suitable choice of basis of Ω_{1}(G_{3}). However now note that due to the ambiguous nature of s_{i} we can shift them by elements of ∧^{2}(x_{1}, …, x_{n}) so without loss of generality the extension element of G_{3} is (s_{1}, …, s_{n}). Note this implies a particular choice of s-basis. Let η be the Bockstein term for this choice of s-basis. Then as we saw in the first sections, when we look at the L.H.S.-spectral sequence for

1 → Ω_{1}(G_{3}) → G_{3} →^{π_{3}} G_{2} → 1

Then

E_{2}^{∗,∗} = ∧^{∗}(t_{1}, …, t_{n}) ⊗ 𝔽_{p}[β(t_{1}), …, β(t_{n})] ⊗ ∧^{∗}(x_{1}, …, x_{n}) ⊗ 𝔽_{p}[s_{1}, …, s_{n}]

where the first two terms are the cohomology of Ω_{1}(G_{3}) and the last two are that of G_{2}. Again d_{2}(β(t_{i})) = d_{2}(x_{i}) = d_{2}(s_{i}) = 0 and d_{2}(t_{i}) = s_{i}. Thus,

E_{3}^{∗,∗} = 𝔽_{p}[β(t_{1}), …, β(t_{n})] ⊗ ∧^{∗}(x_{1}, …, x_{n})

with d_{3}(x_{i}) = 0 and d_{3}(β(t_{i})) = {β(d_{2}(t_{i}))} = {β(s_{i})} in E_{3}. So by what we said before G_{3} has the ΩEP if and only if {β(s_{i})} = 0 for all i. So let us calculate this explicitly. There are elements ξ_{i,l} ∈ ∧^{1}(x_{1}, …, x_{n}) such that:

β(s_{i}) = ∑_{l=1}^{n} ξ_{i,l}s_{l} + β_{2}(s_{i}) = ∑_{l=1}^{n} ξ_{i,l}(0) + β_{2}(s_{i})

where in the last step we used that s_{i} = 0 in E_{3} due to the fact d_{2}(t_{i}) = s_{i}. Thus we get d_{3}(β(t_{i})) = β_{2}(s_{i}) for all i. So that G_{3} defined by the extension element (s_{1}, …, s_{n}) has the ΩEP if and only if β_{2} = 0. As we have argued that the Ω_{1} extensions of G_{2} are exactly those defined by such extension elements, we see that there is an Ω_{1} extension G_{3} of G_{2} such that G_{3} has the ΩEP if and only if there is a choice of s_{1}, …, s_{n} via changing up to adding elements in ∧^{2}(x_{1}, …, x_{n}) so that the corresponding β_{2} = 0, i.e., if and only if [η] = 0. Thus we have shown:

Theorem 2.35. Let {G_{1}, G_{2}} be a uniform tower. Then it extends to a uniform tower of length 3 if and only if 𝔏 = Log(G_{2}) is a Lie algebra. In this case it is possible to extend {G_{1}, G_{2}} to a uniform tower of length 4 if and only if [η] ∈ H^{3}(𝔏; ad) is zero.

2.9. Formulas for B_{2}^{∗}. In this section we will derive explicit formulas for B_{2}^{∗} of the Bockstein Spectral Sequence of a group in LGrp. These formulas will express B_{2}^{∗} in terms of various Lie algebra cohomologies of the Lie algebra corresponding to the group.

Let G ∈ Obj(LGrp) and Log(G) = 𝔏. Let n = dim(𝔏) and assume [η] = 0. Then we have that

H^{∗}(G; 𝔽_{p}) = ∧^{∗}(x_{1}, …, x_{n}) ⊗ 𝔽_{p}[s_{1}, …, s_{n}].

The Bockstein is given by

β(x_{i}) = − ∑_{l<m} c_{lm}^{i}x_{l}x_{m}

and

β(s_{i}) = ∑_{l,m=1}^{n} c_{lm}^{i}s_{l}x_{m}

where c_{lm}^{i} are the structure constants of 𝔏 in a basis {e_{1}, …, e_{n}} implicit in all this. (Note the other Steenrod operations are axiomatically determined so indeed we have determined H^{∗}(G; 𝔽_{p}) as a Steenrod module. So we see the Steenrod module H^{∗}(G; 𝔽_{p}) determines the group in the class of objects LGrp.) Thus we can view H^{∗}(G; 𝔽_{p}) as being bigraded in the same way as E_{∞}^{∗,∗} of the LHS-spectral sequence which we used to compute it. We see that when [η] = 0, the Bockstein β preserves the polynomial degree in this bigrading. Thus the complex (B_{1}^{∗}, β = β_{1}) of the Bockstein spectral sequence splits up as a direct sum of differential complexes, one for each polynomial degree. More precisely, we have

F_{p}[s_{1}, …, s_{n}] = ⊕_{k=0}^{∞}S^{k}

where S^{k} stands for the vector space of homogeneous polynomials of degree k. Thus we have

H^{∗}(G; F_{p}) = ⊕_{k=0}^{∞}(∧^{∗}(x_{1}, …, x_{n}) ⊗ S^{k})

where each ∧^{∗}(x_{1}, …, x_{n}) ⊗ S^{k} becomes a differential complex under β. Thus B_{2}^{∗} is given as the direct sum of the cohomology of each of these complexes. We can identify these complexes with the Koszul complexes used to calculate H^{∗}(𝔏; S^{k}) where S^{k} is the vector space of symmetric multilinear k-forms on 𝔏 given the Lie algebra action:

(u.f)(u_{1}, …, u_{k}) = ∑_{i=1}^{k} f(u_{1}, …, [u_{i}, u], …, u_{k})

for all f ∈ S^{k} and u ∈ 𝔏. To see this note that the direct sum of these Koszul complexes has a natural identification with

⊕_{k=0}^{∞}(∧^{∗}(x_{1}, …, x_{n}) ⊗ S^{k})

and hence H^{∗}(G; F_{p}). In [Pak], it is shown that the sum of the Koszul differentials is a derivation with respect to the algebra structure given, i.e., that of H^{∗}(G; F_{p}). So to show that these are the same as the Bockstein differential it is enough to check that they agree on the generators x_{1}, …, x_{n}, s_{1}, …, s_{n} of H^{∗}(G; F_{p}). This is easy and will be left to the reader (for details see [Pak]). Thus once, grading is taken into consideration one gets the equation:

B_{2}^{∗} = ⊕_{k=0}^{∞}H^{∗−2k}(𝔏, S^{k}) (23)

which holds when [η] = 0. Thus we see that knowing the second term of the Bockstein spectral sequence is equivalent to knowing all the Lie algebra cohomology groups H^{∗}(𝔏; S^{k}). For clarity we state the results of this section in the following theorem.

Theorem 2.36. Let G ∈ Obj(LGrp) with Log(G) = 𝔏 and with [η] = 0. Then B_{2}^{∗} of the Bockstein spectral sequence for G is given by

B_{2}^{∗} = ⊕_{k=0}^{∞}H^{∗−2k}(𝔏; S^{k}).

We will calculate B_{2}^{∗} (at least partially) in some cases later on in this paper.

3. The General Case

3.1. Introduction. Our results so far concern groups G which fit in as G_{2} in some uniform tower

⋯ → G_{n} → ⋯ → G_{2} → G_{1} → 1

We now study the higher groups G_{n} in such a tower. The 𝔽_{p}-cohomology of such groups is determined by theorem 2.32. We will now extend our formulas for the Bockstein to the higher groups G_{n} , n ≥ 3 in such uniform towers.

To do this we will need to use T. Weigel’s Exp-Log correspondence. Let ℤ_{p} denote the p-adic integers.

Definition 3.1. A ℤ_{p}-Lie algebra is a ℤ_{p}-module 𝔏 equipped with a bilinear, alternating product [·, ·] : 𝔏 × 𝔏 → 𝔏 which satisfies the Jacobi Identity.

Definition 3.2. A ℤ_{p}-Lie algebra 𝔏 is powerful if [𝔏, 𝔏] ⊆ p · 𝔏.

Definition 3.3. Ω_{1}(𝔏) = {v ∈ 𝔏 : p · v = 0}

Definition 3.4. A ℤ_{p}-Lie algebra 𝔏 is p-central if [𝔏, Ω_{1}(𝔏)] = 0.

Definition 3.5. CLie is the category of powerful, p-central finite ℤ_{p}-Lie algebras, with morphisms the maps of Lie algebras.

Definition 3.6. CGrp is the category of powerful, p-central p-groups with morphisms the homomorphisms of groups. (see section 2.8 for the relevant definitions.)

Definition 3.7. In analogy to the case with groups we say that 𝔏 ∈ Obj(CLie) has the ΩEP if there exists 𝔏̄ ∈ Obj(CLie) such that 𝔏̄/Ω_{1}(𝔏̄) ≅ 𝔏.

The following theorem was proved in [W3] and stated in [W1]. It generalizes the correspondence between p-adic Lie algebras and uniform pro-p groups (see [DS]) and is based on a careful study of the Baker-Campbell-Hausdorff identity. (e^{A} · e^{B} = e^{C} where C lies in the Lie algebra generated by A and B.)

Theorem 3.8. [W3]: Exp-Log correspondence. Let p ≥ 5. There exist functors Exp : CLie → CGrp and Log : CGrp → CLie which give a natural equivalence of categories. Furthermore if 𝔏 ∈ Obj(CLie) then 𝔏 has the ΩEP if and only if Exp(𝔏) does.

Note that this theorem does not cover the case p = 3.

Definition 3.9. CLie_{k} is the full subcategory of CLie whose objects are the powerful, p-central Lie algebras which are free ℤ/p^{k}ℤ-modules.

Definition 3.10. CGrp_{k} is the full subcategory of CGrp consisting of groups which fit into the kth stage of some uniform tower.

The “exp-log” equivalence above, can easily be shown to restrict to a natural equivalence of CGrp_{k} with CLie_{k} for all k ≥ 1.

Definition 3.11. Brak_{k} is the category whose objects consist of bracket algebras over ℤ/p^{k}ℤ. That is free ℤ/p^{k}ℤ-modules of finite rank B, equipped with a bilinear, alternating map [·, ·] : B × B → B. The morphisms are maps of the underlying modules which preserve the bracket structure.

In [Pak], we construct a shifting function S : CLie_{k} → Brak_{k−1} which is a bijection on the object level and induces certain identifications on the morphisms. We also show that for 𝔏 ∈ Obj(CLie_{k}), then 𝔏 has the ΩEP if and only if S(𝔏) is a Lie algebra. Let Log : CGrp_{k} → Brak_{k−1} be the composition of the functors Log and S. Note this extends the definition of Log : CGrp_{2} → Brak_{1} studied earlier. The facts mentioned above, together with theorem 3.8 allows one to prove the following theorem on uniform towers:

Theorem 3.12. Fix p ≥ 5. Let

G_{k} → ⋯ → G_{2} → G_{1} → 1

be a uniform tower of groups, of length k. Then this tower extends to a uniform tower of length k + 1 if and only if Log(G_{k}) is a Lie algebra.

Proof. The tower extends to a uniform tower of length k + 1 if and only if G_{k} has the ΩEP. By the comments made before, this happens if and only if Log(G_{k}) is a Lie algebra.

Note this extends the k = 2 case proven in theorem 2.10. Let k ≥ 2. Since Log : CGrp_{k} → Brak_{k−1} is a bijection on the object level, we will refer to Exp(B) as the unique group G ∈ Obj(CGrp_{k}) such that Log(G) = B. If one has a morphism ψ ∈ Mor_{Brak_{k−1}}(B_{1}, B_{2}), then one gets a map of groups Ψ : Exp(B_{1}) → Exp(B_{2}) which is unique up to multiplication by a map η : Exp(B_{1}) → Ω_{1}(Exp(B_{2})). (see [Pak].) Although this map is not unique, it induces a unique group homomorphism on the lower levels of the tower. ( Exp(B_{i}) fits into the kth stage of some uniform tower.)

Definition 3.13. For this section, we fix p ≥ 5, a prime. Let LGrp_{k} denote the full subcategory of CGrp_{k} consisting of the objects G ∈ Obj(CGrp_{k}) which have Log(G) a Lie algebra. Let 𝔏 = (W, br) be the reduction of Log(G) to a 𝔽_{p}-Lie algebra.

Then by theorems 2.32 and 3.12, one has that

H^{∗}(G; 𝔽_{p}) ≅ ∧^{∗}(x_{1}, …, x_{n}) ⊗ 𝔽_{p}[s_{1}, …, s_{n}]

where n = dim(𝔏). Recall the exterior algebra comes isomorphically from the lower level in the tower so can be identified with the exterior algebra at the first stage of the tower. Thus by naturality of the Bockstein, we see that

β(x_{i}) = −br^{∗}(x_{i})

which is the same formula as in the second stage of the tower. The polynomial algebra part maps isomorphically to that of Ω_{1}(G) which is isomorphic to the first level of the tower as a group via an iterate of the pth power map. We will use this isomorphism to choose basis from now on without mention - thus maps on the group level will induce the same maps on the x_{i} as they do on the s_{i}. (Up to addition of exterior elements.) As before, we reason that

β(s) = ξs + η

where η is a column vector of elements in ∧^{3}(x_{1}, …, x_{n}), ξ is a n × n matrix of elements in ∧^{1}(x_{1}, …, x_{n}) and s is the column vector of the polynomial generators. As before, because β ∘ β = 0, one can show ξ determines a map of Lie algebras λ : 𝔏 → 𝔤𝔩(𝔏).

Let G_{1}, G_{2} ∈ Obj(LGrp_{k}) and ψ ∈ Mor(Log(G_{1}), Log(G_{2})). Let 𝔏_{i} be the 𝔽_{p}-Lie algebras obtained by reducing Log(G_{i}) for i = 1, 2 and let ψ̄ : 𝔏_{1} → 𝔏_{2} be the map of Lie algebras induced by ψ. Then Exp(ψ) : G_{1} → G_{2} induces a map on the cohomology level which is well defined and induced by pullback via ψ̄ on the exterior generators and which is given by the same formula on the polynomial generators except it is ambiguous up to addition of exterior elements. As before this enables us to prove naturality of the λ maps, that is the various λ maps fit together to form a strong natural self-representation λ on Lie_{k−1}(p).

By theorem 2.25, to show that λ = ad it is enough to show that it is nontrivial on 𝔰𝔩_{2} and 𝔰𝔬_{3}. This follows exactly as it did before by considering the Bockstein spectral sequence and noting that it must converge to zero in positive gradings. Thus if we fix G ∈ Obj(Lie_{k}) and a basis {e_{1}, …, e_{n}} for 𝔏, and let c_{ij}^{k} be the structure constants for 𝔏 with respect to this basis, then we obtain the following formula for the Bockstein as before:

\begin{matrix}β(x_{k}) = & − ∑_{i<j} c_{ij}^{k}x_{i}x_{j} \\ β(s_{k}) = & ∑_{i,j=1}^{n} c_{ij}^{k}s_{i}x_{j} + η_{k} \end{matrix}

Again, one shows η determines a well-defined cohomology class [η] ∈ H^{3}(𝔏; ad) which vanishes if and only if the uniform tower that G is in can be extended two steps above G. We summarize in the following theorem:

Theorem 3.14. Fix p ≥ 5. Let

G_{k+1} → G_{k} → ⋯ → G_{2} → G_{1} → 1

be a uniform tower with k ≥ 2. Let 𝔏 = Log(G_{2}) and let c_{ij}^{k} be the structure constants of 𝔏 with respect to some basis. Then for suitable choices of degree 1 elements x_{1}, …, x_{n} and degree 2 elements s_{1}, …, s_{n}, one has

H^{∗}(G_{k}; 𝔽_{p}) ≅ ∧^{∗}(x_{1}, …, x_{n}) ⊗ 𝔽_{p}[s_{1}, …, s_{n}]

with

\begin{matrix}β(x_{t}) = & − ∑_{i<j}^{n} c_{ij}^{t}x_{i}x_{j} \\ β(s_{t}) = & ∑_{i,j=1}^{n} c_{ij}^{t}s_{i}x_{j} + η_{t} \end{matrix}

for all t = 1, … n. Furthermore η defines a cohomology class [η] ∈ H^{3}(𝔏; ad) which vanishes if and only if there exists a uniform tower

G′_{k+2} → G′_{k+1} → G_{k} → ⋯ → G_{2} → G_{1} → 1

or in other words if and only if Log(G_{k}) (which is a Lie algebra over ℤ/p^{k−1}ℤ), has a lift to a Lie algebra over ℤ/p^{k}ℤ. In this case, one can drop the η terms in the formula for the Bockstein.

Note, for an example of a Lie algebra over ℤ/p^{k−1}ℤ which does not lift to a Lie algebra over ℤ/p^{k}ℤ see [BrP].

Note that the obstruction classes [η] are functionally equivalent to the obstruction classes J_{[·,·]} discussed in [BrP]. Also note that if

H^{∗}(G; 𝔽_{p}) ≅ ∧^{∗}(x_{1}, …, x_{n}) ⊗ 𝔽_{p}[s_{1}, …, s_{n}]

then all the other (non-Bockstein) Steenrod operations are axiomatically determined. So the only difference in the 𝔽_{p}-cohomologies as Steenrod algebras, of the different groups in a uniform tower (except at an end) is in the classes [η]. Thus we conclude the following corollary:

Corollary 3.15. Fix p ≥ 5. Let

G_{k+2} → G_{k+1} → G_{k} → ⋯ → G_{2} → G_{1} → 1

be a uniform tower with k ≥ 2. Then H^{∗}(G_{i}; 𝔽_{p}) are isomorphic Steenrod modules for 2 ≤ i ≤ k.

Finally, one should mention that theorem 3.14 extends to the category of powerful, p-central p-groups with the ΩEP. As mentioned before, there is a natural equivalence Log from this category to the category of powerful, p-central, finite ℤ_{p}-Lie algebras with the ΩEP. The shifting functor S mentioned before (see also [Pak]) maps this category into the category of finite ℤ_{p}-Lie algebras. (Again, the Jacobi identity holds exactly because of the ΩEP as mentioned earlier in this paper.) Unfortunately, the shifting functor does lose a bit of information on the object level which corresponds to an elementary abelian summand in the group/Lie algebra.

Let Log = S ∘ Log, then for a powerful, p-central, p-group G with the ΩEP, one has a corresponding Lie algebra 𝔏 = Log(G) and this induces a Lie algebra over 𝔽_{p}, 𝔏̄ = 𝔏/p𝔏. Due to the loss of the elementary abelian summand, one can have dim(𝔏̄) < dim(Ω_{1}(G)). This is unavoidable, as the Bockstein on the generators of the elementary abelian summand are of a different nature (Bockstein of degree 1 generators is no longer nilpotent for example) but easy to describe as we know what the Bockstein is on elementary abelian p-groups. Taking this into account, one can obtain the following extension of theorem 3.14.

Theorem 3.16. Fix p ≥ 5. Let G be a powerful, p-central p-group with the ΩEP. Let 𝔏 = Log(G), 𝔏̄ = 𝔏/p𝔏 and let c_{ij}^{k} be the structure constants of 𝔏̄ with respect to some basis. Suppose dim(Ω_{1}(G)) = n and dim(𝔏̄) = k. Then for suitable choices of degree 1 elements x_{1}, …, x_{n} and degree 2 elements s_{1}, …, s_{n}, one has

H^{∗}(G; 𝔽_{p}) ≅ ∧^{∗}(x_{1}, …, x_{n}) ⊗ 𝔽_{p}[s_{1}, …, s_{n}]

with

β(x_{t}) = − ∑_{i<j}^{k} c_{ij}^{t}x_{i}x_{j}

β(s_{t}) = ∑_{i,j=1}^{k} c_{ij}^{t}s_{i}x_{j} + η_{t} for t = 1, … k.
β(x_{t}) =s_{t} and β(s_{t}) = 0 for t > k.

Furthermore the η_{t} define a cohomology class [η] ∈ H^{3}(𝔏̄; ad) which vanishes if and only if the Lie algebra 𝔏 has the ΩEP. When this is the case, one can drop the η terms in the formula above.

Here the definition of ΩEP has been extended to finite ℤ_{p}-Lie algebras in general by saying that 𝔏 has the ΩEP if there is a ℤ_{p}-Lie algebra 𝔏̄ such that 𝔏̄/Ω_{1}(𝔏̄) = 𝔏.

3.2. Calculating B_{2}^{∗} in various cases. Recall that there is a natural equivalence between the categories Lie and LGrp. Under this equivalence, Abelian Lie algebras correspond to abelian groups so are not so interesting. Let us work out B_{2}^{∗} for the nonabelian Lie algebra 𝔖 which has basis {x, y} and bracket given by the relation [y, x] = y. Let the corresponding group in LGrp be denoted G(𝔖). This is a group of order p^{4} and exponent p^{2}. Then we have the structure constants c_{y,x}^{y} = 1 and c_{y,x}^{x} = 0. Thus using theorem 3.14, we see that

H^{∗}(G(𝔖); 𝔽_{p}) ≅ ∧^{∗}(x, y) ⊗ 𝔽_{p}[X, Y]

with

\begin{matrix}β(x) = 0 , & β(y) = xy \\ β(X) = 0 , & β(Y) = Y x − Xy \end{matrix}

where we used that the class [η] vanishes as 𝔖 lifts to a Lie algebra over the p-adic integers. Let A be the subalgebra of H^{∗}(G(𝔖); 𝔽_{p}) generated by x, yY^{p−1}, X, Y^{p}. This is isomorphic as graded algebras, to the graded algebra

∧^{∗}(x, z) ⊗ 𝔽_{p}[X, Z]

where deg(x) = 1, deg(z) = 2p − 1, deg(X) = 2, deg(Z) = 2p. It is easy to check that β vanishes on A. Thus there is a well defined map of graded-algebras from A to B_{2}^{∗}. A direct calculation shows that this is an isomorphism. By comparisons (restricting to suitable subgroups of G(𝔖)), it is easy to show then that we can choose z, Z so that β_{2} is given by β_{2}(x) = X, β_{2}(z) = Z. Thus B_{3}^{∗} = 0 for ∗ > 0. Thus we can conclude that

exp(H̄^{∗}(G(𝔖); ℤ)) = p^{2}.

Now let us consider the Lie algebra 𝔰𝔩_{2} with basis {h, x_{+}, x_{−}} with bracket given by [h, x_{+}] = 2x_{+},[h, x_{−}] = −2x_{−} and [x_{+}, x_{−}] = h. Let G(𝔰𝔩_{2}) = Γ̂_{2,2}(p) be the corresponding group. Again we note this Lie algebra lifts to the p-adics so by theorem 3.14 we conclude:

H^{∗}(G(𝔰𝔩_{2}); 𝔽_{p}) ≅ ∧^{∗}(h, x_{+}, x_{−}) ⊗ 𝔽_{p}[H, X_{+}, X_{−}]

with Bockstein given by

\begin{matrix}β(h) = −x_{+}x_{−} , & β(H) = X_{+}x_{−} − X_{−}x_{+} \\ β(x_{+}) = −2hx_{+} , & β(X_{+}) = 2(Hx_{+} − X_{+}h) \\ β(x_{−}) = 2hx_{−} , & β(X_{−}) = −2(Hx_{−} − X_{−}h). \end{matrix}

Using this explicit formula and viewing B_{2}^{∗} as the sum of Lie algebra cohomologies as was shown before, one computes H^{∗}(𝔰𝔩_{2}; 𝔽_{p}) is zero except in dimensions 0 and 3 where it is one dimensional. Also H^{∗}(𝔰𝔩_{2}; S^{1}) = 0 and H^{0}(𝔰𝔩_{2}; S^{2}) is one dimensional generated by the Killing form of 𝔰𝔩_{2}. This implies the following facts: dim(B_{2}^{1}) = dim(B_{2}^{2}) = 0, dim(B_{2}^{3}) = 1 generated by hx_{+}x_{−} and dim(B_{2}^{4}) = 1 generated by the Killing form 8(H^{2} + X_{+}X_{−}). By comparisons, β_{2}(hx_{+}x_{−}) = 0 so that B_{2}^{3} = B_{3}^{3} and thus by necessity (since the Bockstein spectral sequence must eventually converge to zero), B_{2}^{4} = B_{3}^{4}. Thus in particular, B_{3}^{∗} ≠ 0 in positive dimensions. Thus we conclude: exp(H̄^{∗}(G(𝔰𝔩_{2}); ℤ)) > p^{2}. On the other hand, G(𝔰𝔩_{2}) contains G(𝔖)×ℤ/pℤ as a subgroup of index p. Thus by transfer arguments,

exp(H̄^{∗}(G(𝔰𝔩_{2}); ℤ)) ≤ p exp(H̄^{∗}(G(𝔖); ℤ)) = p^{3}.

Thus

exp(H̄^{∗}(G(𝔰𝔩_{2}); ℤ)) = p^{3}.

Note this is neither the order nor the exponent of G(𝔰𝔩_{2}) as G(𝔰𝔩_{2}) has order p^{6} and exponent p^{2}.

The authors would like to thank T. Weigel for many useful discussions.

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