ON COHOMOLOGY OF SPLIT LIE ALGEBRA EXTENSIONS

DIETER DEGRIJSE AND NANSEN PETROSYAN

Abstract. We introduce the notion of compatible actions in the context of split extensions of Lie algebras over a field k. Using compatible actions, we construct new resolutions to compute the cohomology of semi-direct products of Lie algebras and give an alternative way to construct the Hochschild-Serre spectral sequence associated to a split extension. Finally, we describe several instances in which this spectral sequence collapses at the second page and obtain a sharper bound for its length in the finite dimensional case.

1. Introduction

In [6], L. Evens constructed a resolution to compute the cohomology of the semi-direct product H ⋊ G of two groups. This resolution arose by considering a special action of G on a free resolution for H. The construction was later made explicit by T. Brady in [5] where he named it a compatible action. This approach has proven to be very useful for computing the cohomology of certain semi-direct product groups such as crystallographic groups (see for example [1] and [2]).

In this paper, we define the analogue of compatible group actions in the context of Lie algebras over a field k. More concretely, we consider a split extension of Lie algebras 0 → 𝔫 → 𝔤 → 𝔥 → 0 over a field k, a free resolution P → k for 𝔥 and a free resolution F → k for 𝔫. Then we define the notion of compatible action in such a way that, if 𝔥 acts compatibly on F, we can define a 𝔤-module structure on P ⊗_{k} F that turns this complex into a free resolution for 𝔤. Using this fact, we obtain an alternative way to construct the Hochschild-Serre spectral sequence of a split Lie algebra extension, from which we derive the following.

Theorem 1.1. Suppose 0 → 𝔫 → 𝔤 → 𝔥 → 0 is a split extension of Lie algebras. Let M be a 𝔤-module and denote by (E_{r}, d_{r}) the associated Hochschild-Serre spectral sequence. If 𝔥 acts compatibly on a free U(𝔫)-resolution F such that the differential

d^{q−1} : Hom_{𝔫}(F_{q−1}, M) → Hom_{𝔫}(F_{q}, M)

is zero, then d_{r}^{p,q} and d_{r}^{p,q+r−2} are zero for all p and all r ≥ 2.

The accessibility of this construction, of course, depends on the fact whether a particular resolution for 𝔫 admits a compatible action of 𝔥. As it turns out, 𝔥 always acts compatibly on the Chevalley-Eilenberg complex of 𝔫. This allows us to form a practical cochain complex for computing the cohomology of 𝔤. By explicitly constructing compatible actions, we obtain a collapse of the Hochschild-Serre spectral sequence in the following cases.

Date: June 22, 2012

2010 Mathematics Subject Classification. 17B56(primary), 18G60, 18G40(secondary).

Key words and phrases. Lie algebra cohomology, free resolutions, Hochschild-Serre spectral sequence.

The second author was supported by FWO-Flanders Research Fellowship.

Theorem 1.2. Consider the split extension 0 → 𝔫 → 𝔤 → 𝔥 → 0 determined by φ : 𝔥 → Der(𝔫). Let M be a 𝔤-module with a trivial 𝔫-action. Then the Hochschild-Serre spectral sequence associated to this extension with coefficients in M collapses at E_{2} in the following cases

(a) 𝔫 = 𝔫_{1} ⊕ 𝔫_{2}, where 𝔫_{1} is either abelian or free and 𝔫_{2} is either abelian or free;

(b) 𝔫 = 𝔫_{1} ∗𝔫_{2} ∗…∗𝔫_{k}, where each of the 𝔫_{i} is either abelian or free and φ(α)(𝔫_{i}) ⊆ 𝔫_{i} ∀α ∈ 𝔥 and i = 1, …, k.

In [4], D. Barnes showed that the length l of the Hochschild-Serre spectral sequence associated to a split extension of finite dimensional Lie algebras with kernel 𝔫 satisfies l ≤ max {2, dim_{k}(𝔫)} when 𝔫 is nilpotent and acts trivially on the coefficient space. As another corollary of Theorem 1.1, we prove the following generalization of this result.

Theorem 1.3. Suppose 0 → 𝔫 → 𝔤 → 𝔥 → 0 is a split extension of Lie algebras such that dim_{k}(𝔫) = m < ∞. Denote by (E_{r}, d_{r}) the associated Hochschild-Serre spectral sequence with coefficients in a 𝔤-module M. If 𝔫 acts trivially on M, then

(a) d_{r}^{p,m} = 0 for all p and all r ≥ 2;

(b) l ≤ max {2, m};

(c) H^{p}(𝔥, H^{m}(𝔫, M)) ⊕ H^{p+m}(𝔥, M) ⊆ H^{p+m}(𝔤, M) for all p.

2. Definitions, Notations and preliminary results

Suppose R is a ring with unit, and let (A, d^{h}, d^{v}) be a double complex of R-modules. We define the total complex 𝒜 to be the chain complex with 𝒜_{n} = ⨁_{k+l=n} A_{k,l} and differential d defined by d^{h} + d^{v}.

Now, let (P, d) be a chain complex of right R-modules and let (Q, d′) be a chain complex of left R-modules. Then, we define the double complex (B, d^{h}, d^{v}) as B_{p,q} = P_{p} ⊗_{R} Q_{q}

d_{p,q}^{h} : B_{p,q} → B_{p−1,q}, x ⊗ y ↦ d_{p}(x) ⊗ y
d_{p,q}^{v} : B_{p,q} → B_{p,q−1}, x ⊗ y ↦ (−1)^{p}x ⊗ d′_{q}(y).

We define the tensor product of P and Q to be ℬ. In the future we will denote B and ℬ both by P ⊗_{R} Q; the meaning will be apparent from the context.

When (P, d) is a chain complex of left R-modules and (Q, d′) is a cochain complex of left R-modules, we define the double complex (C, d_{h}, d_{v}) as C^{p,q} = Hom_{R}(P_{p}, Q^{q})

d_{h}^{p,q} : C^{p,q} → C^{p+1,q}, f ↦ f ∘ d_{p+1}
d_{v}^{p,q} : C^{p,q} → C^{p,q+1}, f ↦ (−1)^{p}d^{′q} ∘ f.

We denote the total Hom cochain complex of P and Q by 𝒞. Like before, we will abuse notation and denote both C and 𝒞 by Hom_{R}(P, Q).

All Lie algebras we consider are over a fixed field k. Let 𝔤 be a Lie algebra. If M and N are 𝔤-modules then M ⊗_{k} N and Hom_{k}(M, N) naturally become 𝔤-modules in the following way

α(m ⊗ n) = αm ⊗ n + m ⊗ αn, α ∈ 𝔤, m ∈ M, n ∈ N;
(αf)(m) = αf(m) − f(αm), α ∈ 𝔤, m ∈ M, f ∈ Hom_{k}(M, N).

Some useful properties of these 𝔤-module structures are summarized in the following lemma.

Lemma 2.1. There is a natural isomorphism Hom_{k}(M, N)^{𝔤} ≅ Hom_{𝔤}(M, N). Also, the functor Hom_{k}(N, −) : 𝔤-mod → 𝔤-mod is right adjoint to the functor − ⊗_{k} N : 𝔤-mod → 𝔤-mod, which implies that there exists a natural isomorphism

Hom_{𝔤}(M ⊗_{k} N, K) ≅ Hom_{𝔤}(M, Hom_{k}(N, K))

for all 𝔤-modules M, N and K.

Denote by U(𝔤) the universal enveloping algebra of 𝔤. Note that the category of 𝔤modules is naturally isomorphic to the category of U(𝔤)-modules, so we will identify them without mentioning. The cohomology of 𝔤 with coefficients in the 𝔤-module M is defined as H^{∗}(𝔤, M) = Ext_{U(𝔤)}^{∗}(k, M). Hence, H^{∗}(𝔤, M) can be computed by taking the cohomology of Hom_{𝔤}(F, M), where F is any free U(𝔤)-resolution of k. For details on homological algebra and the cohomology of Lie algebras, we refer the reader to [8] and [10].

Lemma 2.2. Let 0 → 𝔫 → 𝔤 ⟶^{π} 𝔥 → 0 be short exact sequence of Lie algebras. If K, N are 𝔤-modules such that 𝔫 acts trivially on K, then there is a natural isomorphism

Hom_{𝔤}(K, N) ≅ Hom_{𝔥}(K, N^{𝔫}).

In particular, we have a natural isomorphism of functors −^{𝔤} ≅ −^{𝔥} ∘−^{𝔫}, where we consider −^{𝔫} as a functor from 𝔤-mod to 𝔥-mod.

3. Compatible Actions

We are especially interested in split short exact sequences of Lie algebras over a field k

(1)  \xymatrix{0 \ar[r] & 𝔫 \ar[r] & 𝔤 \ar[r]^{π} & 𝔥 \ar[l] \ar[r] & 0} .

There is a Lie algebra homomorphism φ : 𝔥 → Der(𝔫), where Der(𝔫) is the derivation algebra of 𝔫. Using φ, we can write 𝔤 as a semi-direct product 𝔤 = 𝔫 ⋊_{φ} 𝔥. Viewed this way, multiplication in 𝔤 is given by

[(s, α), (t, β)] = ([s, t] + φ(α)(t) − φ(β)(s), [α, β]), ∀α, β ∈ 𝔥, s, t ∈ 𝔫.

In what follows, we will drop φ from our notation and write φ(α)(t) as α(t) for all α ∈ 𝔥 and t ∈ 𝔫. Given a 𝔤-module M, we will construct a new resolution to compute H^{∗}(𝔤, M). Our result will depend on the existence of what is called a compatible action.

Definition 3.1. Suppose ε : F → k is a free resolution of k over U(𝔫). Let ℭ(F) be the set of chain maps from F to itself that extend the zero map on k. It is an associative k-algebra under composition and hence it can be given the standard Lie algebra structure. We say 𝔥 acts compatibly on F, if there exists a Lie algebra homomorphism Θ : 𝔥 → ℭ(F) : α ↦ \underline{α}, such that

(2)  α(s)f = \underline{α}(sf) − s\underline{α}(f)

for all α ∈ 𝔥, s ∈ 𝔫 and f ∈ F_{∗}.

Given an 𝔥-module M, we can use the projection map π : 𝔤 → 𝔥 to turn M into a 𝔤-module. Moreover, a U(𝔥)-resolution of k inflates to a U(𝔤)-resolution of k. However, since the projection of 𝔤 onto 𝔫 is not a Lie algebra homomorphism, there is no obvious way of extending a U(𝔫)-resolution to a U(𝔤)-resolution. This is where compatible actions come into play.

Lemma 3.2. Suppose there is a compatible action of 𝔥 on a U(𝔫)-resolution ε : F → k. Let (s, α) ∈ 𝔤 for s ∈ 𝔫, α ∈ 𝔥, and f ∈ F_{∗}, then

(3)  (s, α)f = sf + \underline{α}(f)

turns F → k into a resolution of U(𝔤)-modules.

Proof. For each n, denote by F_{n} the n^{th}-module of F. By definition of compatible action, the action in (3) turns F_{n} into a 𝔤-module.

To see that the differentials of F are 𝔤-module homomorphisms, we use the fact that \underline{α} is a chain map for each α ∈ 𝔥. Let f ∈ F_{n} and (s, α) ∈ 𝔤, then

d((s, α)f) = d(sf + \underline{α}(f)) = sd(f) + \underline{α}(d(f)) = (s, α)d(f).

Finally, the augmentation ε : F_{0} → k becomes a 𝔤-module map (give k trivial 𝔤-module structure) because \underline{α} extends the zero map on k for each α ∈ 𝔥. Let f ∈ F_{0} and (s, α) ∈ 𝔤. Then, we have

ε((s, α)f) = sε(f) + ε(\underline{α}(f)) = 0 = (s, α)ε(f).
□

Next, we consider a free U(𝔫)-resolution ε_{F} : F → k and assume that it admits a compatible action of 𝔥. Using Lemma 3.2, we inflate ε_{F} : F → k into a (not necessarily free) U(𝔤)-resolution of k. Also, we consider a free U(𝔥)-resolution ε_{P} : P → k of k and turn it into a U(𝔤)-resolution of k, using the projection map π. The complex P ⊗_{k} F now turns out to be a free resolution of U(𝔤)-modules. To summarize, we have

Lemma 3.3. The complex ε_{P} ⊗ε_{F} : P ⊗_{k} F → k is a free U(𝔤)-resolution, with the action of U(𝔤) on P ⊗_{k} F induced by

(s, α)(p ⊗ f) := αp ⊗ f + p ⊗ (sf + \underline{α}(f))

for each (s, α) ∈ 𝔤, p ∈ P_{∗}, and f ∈ F_{∗}.

Proof. From the Künneth formula for tensor products, it follows that ε_{P} ⊗ε_{F} : P ⊗_{k}F → k is a U(𝔤)-resolution of k.

The n^{th}-module of P ⊗_{k} F is given by ⨁_{p+q=n} P_{q} ⊗_{k} F_{q}, and we need to show that this is a free U(𝔤)-module. Because P consists of free U(𝔥)-modules, it suffices to show that U(𝔥) ⊗_{k} F_{q} is a free U(𝔤)-module for every q. Furthermore, it follows from tensor identities that the 𝔤-modules U(𝔥) ⊗_{k} F_{q} and U(𝔤) ⊗_{U(𝔫)} F_{q} are isomorphic, where the 𝔤-module structure on U(𝔤) ⊗_{U(𝔫)} F is given by multiplication on the left in U(𝔤). Hence, we see that

U(𝔥) ⊗_{k} F_{q} ≅ U(𝔤) ⊗_{U(𝔫)} F_{q} ≅ U(𝔤) ⊗_{U(𝔫)} ( ⊕_{i∈I} U(𝔫)) ≅ ⊕_{i∈I}U(𝔤).
□

4. Constructing compatible actions

We will first show that compatible actions always exist for the Chevalley-Eilenberg complex V (𝔫) of 𝔫:

… → U(𝔫) ⊗_{k} Λ^{p}(𝔫) ⟶^{d_{p}} … ⟶^{d_{2}} U(𝔫) ⊗_{k} Λ^{1}(𝔫) ⟶^{d_{1}} U(𝔫) ⊗ Λ^{0}(𝔫) ⟶^{ε} k → 0.

where Λ^{p}(𝔤) denotes the p-th exterior product of 𝔤, ε is the usual augmentation map, and d_{1} : U(𝔫) ⊗_{k} 𝔫 → U(𝔫) is the product map d_{1}(u ⊗ x) = ux. For p ≥ 2, and u ⊗ x_{1} ∧ … ∧ x_{p} ∈ V_{p}(𝔫), (u ∈ U(𝔫), x_{i} ∈ 𝔫) the boundary map is given by

d_{p}(u ⊗ x_{1} ∧ … ∧ x_{p}) = ∑_{i=1}^{p}(−1)^{i+1}ux_{i}x_{1} ∧ … ∧ x̂_{i} ∧ … ∧ x_{p}+

∑_{i<j}(−1)^{i+j}u ⊗ [x_{i}, x_{j}] ∧ x_{1} ∧ … ∧ x̂_{i} ∧ … ∧ x̂_{j} ∧ … ∧ x_{p}.

Proposition 4.1. Given the split extension (1), the maps

\underline{α} : U(𝔫) ⊗_{k} Λ^{p}(𝔫) → U(𝔫) ⊗_{k} Λ^{p}(𝔫) :
1 ⊗ x_{1} ∧ … ∧ x_{p} ↦ ∑_{j=1}^{p} 1 ⊗ x_{1} ∧ … ∧ α(x_{j}) ∧ … ∧ x_{p},

y_{1} … y_{m} ⊗ x_{1} ∧ … ∧ x_{p} ↦ ∑_{j=1}^{m} y_{1} … α(y_{j}) … y_{m} ⊗ x_{1} ∧ … ∧ x_{p},

+ ∑_{j=1}^{p} y_{1} … y_{m} ⊗ x_{1} ∧ … ∧ α(x_{j}) ∧ … ∧ x_{p}

for all α ∈ 𝔥, define a compatible action of 𝔥 on the Chevalley-Eilenberg complex of 𝔫. (If p = 0, then the second big sum disappears.)

Proof. Let us first show that for each α ∈ 𝔥, \underline{α} is an augmentation preserving chain map. By simple computations, this reduces to showing that d ∘ \underline{α}(1 ⊗ x_{1} ∧ … ∧ x_{p}) = \underline{α} ∘ d(1 ⊗ x_{1} ∧ … ∧ x_{p}), for all p. First, we compute the left hand side (L).

(L) = ∑_{j=1}^{p} d(1 ⊗ x_{1} ∧ … α(x_{j}) … ∧ x_{p})
= ∑_{j=1}^{p}(−1)^{j+1}α(x_{j}) ⊗ x_{1} ∧ … ∧ x̂_{j} ∧ … ∧ x_{p}

+ ∑_{\begin{matrix}l,j=1 \\ l≠j \end{matrix}}^{p}(−1)^{l+1}x_{l} ⊗ x_{1} ∧ … ∧ x̂_{l} ∧ … ∧ α(x_{j}) ∧ … ∧ x_{p}

+ ∑_{l>j}^{p}(−1)^{l+j} ⊗ [α(x_{j}), x_{l}] ∧ x_{1} ∧ … ∧ x̂_{j} ∧ … ∧ x̂_{l} ∧ … ∧ x_{p}

+ ∑_{j>l}^{p}(−1)^{l+j} ⊗ [x_{l}, α(x_{j})] ∧ x_{1} ∧ … ∧ x̂_{l} ∧ … ∧ x̂_{j} ∧ … ∧ x_{p}

+ ∑_{j=1}^{p} ∑_{\begin{matrix}l>k \\ l≠j≠k \end{matrix}}^{p}(−1)^{l+k} ⊗ [x_{k}, x_{l}] ∧ x_{1} ∧ … ∧ x̂_{k} ∧ … ∧ x̂_{l} ∧ … ∧ α(x_{j}) ∧ … ∧ x_{p}.

Since α acts as a derivation, we have α([x_{l}, x_{j}]) = [α(x_{l}), x_{j}] + [x_{l}, α(x_{j})]. So, continuing with the equality, we find

(L) = ∑_{j=1}^{p}(−1)^{j+1}α(x_{j}) ⊗ x_{1} ∧ … ∧ x̂_{j} ∧ … ∧ x_{p}

+ ∑_{\begin{matrix}l,j=1 \\ l≠j \end{matrix}}^{p}(−1)^{j+1}x_{j} ⊗ x_{1} ∧ … ∧ x̂_{j} ∧ … ∧ α(x_{l}) ∧ … ∧ x_{p}

+ ∑_{j>l}^{p}(−1)^{l+j}1 ⊗ α([x_{l}, x_{j}]) ∧ x_{1} ∧ … ∧ x̂_{l} ∧ … ∧ x̂_{j} ∧ … ∧ x_{p}

+ ∑_{j=1}^{p} ∑_{\begin{matrix}l>k \\ l≠j≠k \end{matrix}}^{p}(−1)^{l+k} ⊗ [x_{k}, x_{l}] ∧ x_{1} ∧ … ∧ x̂_{k} ∧ … ∧ x̂_{l} ∧ … ∧ α(x_{j}) ∧ … ∧ x_{p}.

Meanwhile, the right hand side (R) is

(R) = ∑_{j=1}^{p}(−1)^{j+1}\underline{α}(x_{j} ⊗ x_{1} ∧ … ∧ x̂_{j} ∧ … ∧ x_{p})

+ ∑_{j>l}^{p}(−1)^{l+j}\underline{α}(1 ⊗ [x_{l}, x_{j}] ∧ x_{1} ∧ … ∧ x̂_{l} ∧ … ∧ x̂_{j} ∧ … ∧ x_{p})

= ∑_{j=1}^{p}(−1)^{j+1}α(x_{j}) ⊗ x_{1} ∧ … ∧ x̂_{j} ∧ … ∧ x_{p}

+ ∑_{\begin{matrix}l,j=1 \\ l≠j \end{matrix}}^{p}(−1)^{j+1}x_{j} ⊗ x_{1} ∧ … ∧ x̂_{j} ∧ … ∧ α(x_{l}) ∧ … ∧ x_{p}

+ ∑_{j<l}^{p}(−1)^{l+j}\underline{α}(1 ⊗ [x_{l}, x_{j}] ∧ x_{1} ∧ … ∧ x̂_{l} ∧ … ∧ x̂_{j} ∧ … ∧ x_{p}).

Now, using the definition of \underline{α}, we see that this is the same expression as before.

Next, straightforward computations confirm that the map Θ : 𝔥 → ℭ(V (𝔫)) is a Lie algebra homomorphism.

It is left to check condition (2). Suppose y_{1}y_{2} … y_{m} ∈ U(𝔫), x_{1} ∧ … ∧ x_{p} ∈ Λ^{p}(𝔫) and x ∈ 𝔫. Then,

\underline{α}(xy_{1}y_{2} … y_{m} ⊗ x_{1} ∧ … ∧ x_{p}) = ∑_{j=1}^{m} xy_{1} … α(y_{j}) … y_{m} ⊗ x_{1} ∧ … ∧ x_{p}
+α(x)y_{1}y_{2} … y_{m} ⊗ x_{1} ∧ … ∧ x_{p}
+xy_{1}y_{2} … y_{m}\underline{α}(1 ⊗ x_{1} ∧ … ∧ x_{p})
= α(x)y_{1} … y_{m} ⊗ x_{1} ∧ … ∧ x_{p} +
x\underline{α}(y_{1} … y_{m} ⊗ x_{1} ∧ … ∧ x_{p}).

This shows that condition (2) is satisfied. Now, using the definition of \underline{α}, we see that this is the same expression as before. We conclude that the maps \underline{α} indeed define a compatible action of 𝔥 on the Chevalley-Eilenberg complex of 𝔫. □

Next, we give four simple but useful lemmas that allow us to construct new compatible actions from already existing ones. The proofs of the first three lemmas are straightforward.

Lemma 4.2. Let 𝔥_{1} → Der(𝔫) be a Lie algebra homomorphism and suppose 𝔥_{1} acts compatibly on a free U(𝔫)-resolution ε : F → k. If ϕ : 𝔥_{2} → 𝔥_{1} is a Lie algebra homomorphism, then \underline{α}(f) = \underline{(ϕ(α))}(f) defines a compatible action of 𝔥_{2} on F.

Lemma 4.3. Suppose that, for i = 1, 2, we have a Lie algebra homomorphism φ_{i} : 𝔥 → Der(𝔫_{i}) such that 𝔥 acts compatibly on a free U(𝔫_{i})-resolution ε_{i} : F_{i} → k. Then, considering the homomorphism φ : 𝔥 → Der(𝔫_{1}) ⊕ Der(𝔫_{2}) ↪ Der(𝔫_{1} ⊕ 𝔫_{2}), we obtain a compatible action of 𝔥 on the free U(𝔫_{1} ⊕ 𝔫_{2})-resolution ε_{1} ⊗ ε_{2} : F_{1} ⊗_{k} F_{2} → k given by \underline{α}(f_{1} ⊗ f_{2}) := \underline{α}(f_{1}) ⊗ f_{2} + f_{2} ⊗ \underline{α}(f_{2}).

Lemma 4.4. Let 𝔥 = 𝔥_{1}⋊_{ρ}𝔥_{2} and let φ : 𝔥 → Der(𝔫) be a Lie algebra homomorphism. For each i = 1, 2, suppose 𝔥_{i} acts compatibly through φ on the free U(𝔫)-resolution ε : F → k. If \underline{ρ(α_{2})(α_{1})} = \underline{α_{2}} ∘ \underline{α_{1}} − \underline{α_{1}} ∘ \underline{α_{2}} for all (α_{1}, α_{2}) ∈ 𝔥, then \underline{(α_{1}, α_{2})}(f) = \underline{α_{1}}(f) + \underline{α_{2}}(f) defines a compatible action of 𝔥 on F.

Lemma 4.5. Suppose 𝔫_{1} and 𝔫_{2} are two Lie algebras and consider a Lie algebra homomorphism φ : 𝔥 → Der(𝔫_{1} ∗ 𝔫_{2}) such that for every α ∈ 𝔥, we have φ(α)(𝔫_{i}) ⊂ 𝔫_{i} for i = 1, 2. Then

… → U(𝔫_{1} ∗ 𝔫_{2}) ⊗ (Λ^{p}(𝔫_{1}) ⊕ Λ^{p}(𝔫_{2})) → … → U(𝔫_{1} ∗ 𝔫_{2}) → k → 0.

is a free U(𝔫_{1} ∗ 𝔫_{2})-resolution of k that allows a compatible action of 𝔥.

Proof. Given two Lie algebras 𝔫_{1} and 𝔫_{2}, we can consider their free product 𝔫_{1} ∗ 𝔫_{2}. If J_{i} is the augmentation ideal of 𝔫_{i} and J is the augmentation ideal of 𝔫_{1} ∗ 𝔫_{2}, then

J ≅ (U(𝔫_{1} ∗ 𝔫_{2}) ⊗_{U(𝔫_{1})} J_{1}) ⊕ (U(𝔫_{1} ∗ 𝔫_{2}) ⊗_{U(𝔫_{2})} J_{2})

as left U(𝔫_{1} ∗ 𝔫_{2})-modules. Denote by V_{∗}(𝔫_{i}) the Chevalley-Eilenberg resolution of 𝔫_{i}, for i = 1, 2. Then

(4)  … → V_{p}(𝔫_{i}) → V_{p−1}(𝔫_{i}) → . . . . . . V_{1}(𝔫_{i}) → J_{i} → 0

is a free U(𝔫_{i})-resolution of J_{i}, for i = 1, 2. Since U(𝔫_{1} ∗ 𝔫_{2}) is a free U(𝔫_{i})-module, applying U(𝔫_{1} ∗ 𝔫_{2}) ⊗_{U(𝔫_{i})} − to (4) yields an exact U(𝔫_{1} ∗ 𝔫_{2})-complex

(5) … → U(𝔫_{1} ∗ 𝔫_{2}) ⊗ Λ^{p}(𝔫_{i}) → U(𝔫_{1} ∗ 𝔫_{2}) ⊗ Λ^{p−1}(𝔫_{i}) → … → U(𝔫_{1} ∗ 𝔫_{2}) ⊗_{U(𝔫_{i})} J_{i} → 0

for i = 1, 2. If we now take the direct sum of (5) for i = 1, 2, we obtain a free U(𝔫_{1} ∗ 𝔫_{2})- resolution of J that we can extend to a free U(𝔫_{1} ∗ 𝔫_{2})-resolution of k, i.e.

(6)  … → U(𝔫_{1} ∗ 𝔫_{2}) ⊗ (Λ^{p}(𝔫_{1}) ⊕ Λ^{p}(𝔫_{2})) → … → U(𝔫_{1} ∗ 𝔫_{2}) → k → 0.

Now suppose we have a Lie algebra homomorphism φ : 𝔥 → Der(𝔫_{1} ∗ 𝔫_{2}) such that for every α ∈ 𝔥, we have φ(α)(𝔫_{i}) ⊂ 𝔫_{i} for i = 1, 2. Then 𝔥 acts compatibly on (6). Indeed, the compatible action on the n-th module of (6) can be defined on each of the two direct summands U(𝔫_{1} ∗ 𝔫_{2}) ⊗ Λ^{p}(𝔫_{i}) as in Proposition 4.1. □

Remark 4.6. Note that this lemma can easily be generalized to free products of more that two factors. Also, if one of the factors, say 𝔫_{2}, is free then we can replace the Chevalley-Eilenberg complex of 𝔫_{2} by a resolution of the form (7).

In the next examples, we will apply these lemmas to construct several useful compatible actions.

Example 4.7. Let 𝔣_{m} be the free Lie algebra on m generators {x_{1}, …, x_{m}} = X. Then k{X}, the free k-algebra on X, is the universal enveloping algebra of 𝔣_{m}. Since the augmentation ideal J of k{X} can be seen as an m-dimensional free k{X}-module,

(7)  0 → J → k{X} → k → 0

is a free U(𝔣_{𝔪})-resolution of k. Now, consider the universal split extension 𝔣_{m} ⋊ Der(𝔣_{𝔪}) and take α ∈ Der(𝔣_{𝔪}). Then one can easily check that the k-linear map

\underline{α} : k\{X\} → k\{X\} : \{ \begin{matrix}r ∈ k & ↦ & 0 \\ x_{i_{1}}x_{i_{2}} … x_{i_{p}} & ↦ & ∑_{j=1}^{p} x_{i_{1}} … α(x_{i_{j}}) … x_{i_{p}} \end{matrix}

induces a compatible action of Der(𝔣_{𝔪}) on (7). Hence, by Lemma 4.2, every split extension 𝔣_{n} ⋊ 𝔥 allows a compatible action of 𝔥 on (7).

Example 4.8. Let 𝔣_{1} be the free Lie algebra on m generators {x_{1}, …, x_{m}} = X and take 𝔣_{2} to be the free Lie algebra on n generators {y_{1}, …, y_{n}} = Y. Now consider (𝔣_{1} ⊕ 𝔣_{2}) ⋊ Der(𝔣_{1} ⊕𝔣_{2}). Since Der(𝔣_{m} ⊕𝔣_{n}) = Der(𝔣_{m})⊕Der(𝔣_{n}), we can use the previous example and Lemma 4.3 to obtain a compatible action of Der(𝔣_{m} ⊕ 𝔣_{n}) on a free U(𝔣_{1} ⊕ 𝔣_{2})-resolution of k. Using Lemma 4.2, we see that any split extension with kernel 𝔣_{1} ⊕ 𝔣_{2} admits a compatible action of this form.

Example 4.9. Let 𝔣 be the free Lie algebra on m generators {x_{1}, …, x_{m}} = X and let k^{n} be the n-dimensional abelian Lie algebra with k-basis {t_{1}, …, t_{n}}. Recall that the universal enveloping algebra of k^{n} equals the polynomial ring in n variables k[t_{1}, …, t_{n}] and that we have a split short exact sequence

0 → Der(𝔣, k^{n}) → Der(𝔣 ⊕ k^{n}) → Der(𝔣) ⊕ Der(k^{n}) → 0.

Here, Der(𝔣, k^{n}) denotes the abelian Lie algebra of all k-linear maps from 𝔣 to k^{n} that map [𝔣, 𝔣] to zero.

Now, denote by F_{1} the free U(𝔣)-resolution (7) and by F_{2} the Chevalley-Eilenberg resolution of k^{n}. By Example 4.7, we have a compatible action of Der(𝔣) on F_{1} and by Proposition 4.1 we have a compatible action of Der(k^{n}) on F_{2}. Hence, it follows from Lemma 4.3 that we have a compatible action of Der(𝔣) ⊕ Der(k^{n}) on F = F_{1} ⊗ F_{2}. Next we will construct a compatible action of Der(𝔣, k^{n}) on F. To simplify our notation, we will first rewrite the complex F. Define

A_{0} = k,
A_{p} = Λ^{p}(k^{n}) ⊕ (Λ^{p−1}(k^{n}) ⊗ ⟨x_{1}, …, x_{m}⟩) for 1 ≤ p ≤ n,
A_{n+1} = Λ^{n}(k^{n}) ⊗ ⟨x_{1}, …, x_{m}⟩,

where ⟨x_{1}, …, x_{m}⟩ is the m-dimensional vector space with basis X. Then one can check that F_{p} = U(𝔣 ⊕ k^{n}) ⊗ A_{p} for all p ∈ {0, …, n + 1}, with differentials given by

d_{1} : F_{1} → F_{0} : w ⊗ (t_{i}, x_{k}) ↦ w(x_{k}, t_{i})
d_{p} : F_{p} → F_{p−1} : w ⊗ (t_{i_{1}} ∧ … ∧ t_{i_{p}}, t_{j_{1}} ∧ … ∧ t_{j_{p−1}} ⊗ x_{k}) ↦

∑_{r=1}^{p}(−1)^{r+1}w(0, t_{i_{r}}) ⊗ (t_{i_{1}} ∧ … ∧ t̂_{i_{r}} ∧ … ∧ t_{i_{p}}, 0)

+ ∑_{s=1}^{p−1}(−1)^{s}w(0, t_{j_{s}}) ⊗ (0, t_{j_{1}} ∧ … ∧ t̂_{j_{s}} ∧ … ∧ t_{j_{p−1}} ⊗ x_{k})

+w(x_{k}, 0) ⊗ (t_{j_{1}} ∧ … ∧ t_{j_{p−1}}, 0) for all p ∈ {2, …, n} and

d_{n+1} : F_{n+1} → F_{n} : w ⊗ (t_{1} ∧ … ∧ t_{n} ⊗ x_{k}) ↦

∑_{j=1}^{n}(−1)^{j}w(0, t_{j}) ⊗ (0, t_{1} ∧ … ∧ t̂_{j} ∧ … ∧ t_{n} ⊗ x_{k})

+w(x_{k}, 0) ⊗ (t_{1} ∧ … ∧ t_{n}, 0).

Now, take α ∈ Der(𝔣, k^{n}) ⊆ Der(𝔣 ⊕ k^{n}) and a_{i} ∈ U(𝔣 ⊕ k^{n}), then one can check that the maps

\underline{α} : F_{0} → F_{0} : a_{1}a_{2} … a_{r} ↦ ∑_{s=1}^{r} a_{1} … α(a_{s}) … a_{r}
\underline{α} : F_{p} → F_{p} : a_{1}a_{2} … a_{r} ⊗ (t_{i_{1}} ∧ … ∧ t_{i_{p}}, t_{j_{1}} ∧ … ∧ t_{j_{p−1}} ⊗ x_{k}) ↦

∑_{s=1}^{r} a_{1} … α(a_{s}) … a_{r} ⊗ (t_{i_{1}} ∧ … ∧ t_{i_{p}}, t_{j_{1}} ∧ … ∧ t_{j_{p−1}} ⊗ x_{k})

+(−1)^{p−1}a_{1} … a_{r} ⊗ (t_{j_{1}} ∧ … ∧ t_{j_{p−1}} ∧ α(x_{k}), 0)

\underline{α} : F_{n+1} → F_{n+1} : a_{1}a_{2} … a_{r} ⊗ (t_{1} ∧ … ∧ t_{n} ⊗ x_{k}) ↦

∑_{s=1}^{r} a_{1} … α(a_{s}) … a_{r} ⊗ (t_{1} ∧ … ∧ t_{n} ⊗ x_{k})

define a compatible action of Der(𝔣, k^{n}) on F. Furthermore, one can verify that

\underline{φ_{2} ∘ α} − \underline{α ∘ φ_{1}} = \underline{(φ_{1}, φ_{2})} ∘ \underline{α} − \underline{α} ∘ \underline{(φ_{1}, φ_{2})}

for all (φ_{1}, φ_{2}) ∈ Der(𝔣) ⊕ Der(k^{n}) and for all α ∈ Der(𝔣, k^{n}) which means that the action of Der(𝔣, k^{n}) is compatible with the action of Der(𝔣) ⊕ Der(k^{n}) in the sense of Lemma 4.4. It now follows from Lemma 4.4 that we have a compatible action of Der(𝔣 ⊕ k^{n}) on F. Finally, Lemma 4.2 implies that every split extension with kernel 𝔣 ⊕ k^{n} admits a compatible action of this form.

5. The Hochschild-Serre spectral sequence of a split extension

Recall that a short exact sequence of Lie algebras

(8)  0 → 𝔫 → 𝔤 ⟶^{π} 𝔥 → 0

and a 𝔤-module M give rise to a Hochschild-Serre spectral sequence. For a general treatment of spectral sequences we refer the reader to [9] and [10]. The Hochschild-Serre spectral sequence for Lie algebra extensions is discussed in [3] and [7].

When the extension (8) splits, we propose a modification to the construction of the Hochschild-Serre spectral sequence.

Proposition 5.1. Let 0 → 𝔫 → 𝔤 → 𝔥 → 0 be a split extension of Lie algebras and let M be a 𝔤-module. If ε_{P} : P → k is a free U(𝔥)-resolution and ε_{F} : F → k is a free U(𝔫)-resolution that allows a compatible action of 𝔥, then this action defines a 𝔤-module structure on F such that,

H^{n}(𝔤, M) = H^{n}(Hom_{𝔥}(P, Hom_{𝔫}(F, M)))

for each n.

Proof. According to Lemma 3.3, ε_{P} ⊗ε_{F} : P ⊗_{k}F → k is a free U(𝔤)-resolution. Therefore,

H^{∗}(𝔤, M) = H^{∗}(Hom_{𝔤}(P ⊗_{k} F, M)).

Also, by Lemma 2.1, we have

Hom_{𝔤}(P ⊗_{k} F, M) ≅ Hom_{𝔤}(P, Hom_{k}(F, M)).

Furthermore, since 𝔫 acts trivially on P_{p} for each p, it follows from Lemmas 2.1 and 2.2 that

Hom_{𝔤}(P_{q}, Hom_{k}(F_{q}, M)) = Hom_{𝔥}(P_{p}, Hom_{𝔫}(F_{q}, M))

for all p and q. We conclude that H^{∗}(𝔤, M) can be calculated by taking the cohomology of Hom_{𝔥}(P, Hom_{𝔫}(F, M)). □

Filtering by columns, we can obtain a canonically bounded filtration of the (total) Hom cochain complex Hom_{𝔥}(P, Hom_{𝔫}(F, M)). By constructing the spectral sequence associated to this filtration and using the proposition above, we obtain a convergent first quadrant spectral sequence

(9)  E_{2}^{p,q} = H^{p}(𝔥, H^{q}(𝔫, M)) ⇒ H^{p+q}(𝔤, M).

Moreover, it is not difficult to see that this spectral sequence coincides with the Hochschild-Serre spectral sequence from the second page onward. We will use this different construction of the Hochschild-Serre spectral sequence to prove a generalization of Theorem 2 from [4], but first we need a lemma.

Lemma 5.2. Suppose (C, d_{h}, d_{v}) is a first quadrant double complex with the vertical differential d_{v}^{p+1,q−1} : C^{p+1,q−1} → C^{p+1,q} zero for some p and q. Then the differentials d_{r}^{p,q} and d_{r}^{p−r+2,q+r−2}, from the convergent first quadrant spectral sequence

IE_{2}^{p,q} = H_{h}^{p}H_{v}^{q}(C) ⇒ H^{p+q}(𝒞),

obtained by filtering C columnwise, are zero for all r ≥ 2.

Proof. Recall that 𝒞 is the cochain complex with 𝒞^{n} = ⨁_{k+l=n} C^{k,l}, and the differential d is defined by d_{h} +d_{v}. The filtration of 𝒞 is given by F^{p}𝒞^{n} = ⨁\begin{matrix}k+l=n \\ k≥p \end{matrix} C^{k,l}. By definition

we have E_{r}^{p,q} = Z_{r}^{p,q}/(Z_{r−1}^{p+1,q−1} + B_{r−1}^{p,q}), with

Z_{r}^{p,q} = F^{p}𝒞^{p+q} ∩ d^{−1}(F^{p+r}𝒞^{p+q+1}),
B_{r}^{p,q} = F^{p}𝒞^{p+q} ∩ d(F^{p−r}𝒞^{p+q−1}).

Also, the differentials d_{r}^{p,q} : E_{r}^{p,q} → E_{r}^{p+r,q−r+1} are induced by the restriction of d to Z_{r}^{p,q}.

Now, let [x] ∈ E_{r}^{p,q} where x ∈ Z_{r}^{p,q}. We can write x = f + x′ with f ∈ C^{p,q} and x′ ∈ F^{p+1}𝒞^{p+q}. Since d_{v}^{p+1,q−1} = 0, we have d(x) = d(x′) (if r ≥ 2). This means that d(x) ∈ F^{p+r}𝒞^{p+q+1} ∩d(F^{p+1}𝒞^{p+q}) = B_{r−1}^{p+r,q−r+1} showing that d_{r}^{p,q}([x]) = 0. Since [x] and r are arbitrary, we conclude that d_{r}^{p,q} = 0 for all r ≥ 0.

Similarly, take [x] ∈ E_{r}^{p−r+2,q+r−2} where x ∈ Z_{r}^{p−r+2,q+r−2} ⊂ F^{p−r+2}𝒞^{p+q}. Then d_{r}^{p−r+2,q+r−2}([x]) = [d(x)] ∈ E_{r}^{p+2,q−1}. We will show that d(x) ∈ B_{r−1}^{p+2,q−1}. Denote by x′ the image of x under the projection of F^{p−r+2}𝒞^{p+q} onto F^{p+1}𝒞^{p+q}. Because d_{v}^{p+1,q−1} = 0, one can easily verify that d(x) = d(x′). But this implies that d(x) ∈ B_{r−1}^{p+2,q−1}, because F^{p+1}𝒞^{p+q} ⊂ F^{p−r+3}𝒞^{p+q} for r ≥ 2. By definition of E_{r}^{p+2,q−1}, this means that d_{r}^{p−r+2,q+r−2}([x]) = 0. Since [x] and r are arbitrary, we conclude that d_{r}^{p−r+2,q+r−2} = 0 for all r ≥ 0. □

Theorem 5.3. Suppose 0 → 𝔫 → 𝔤 → 𝔥 → 0 is a split extension of Lie algebras. Let M be a 𝔤-module and denote by (E_{r}, d_{r}) the associated Hochschild-Serre spectral sequence. If 𝔥 acts compatibly on a free U(𝔫)-resolution F such that the differential

d^{q−1} : Hom_{𝔫}(F_{q−1}, M) → Hom_{𝔫}(F_{q}, M)

is zero, then d_{r}^{p,q} and d_{r}^{p,q+r−2} are zero for all p and all r ≥ 2.

Proof. If d^{q−1} : Hom_{𝔫}(F_{q−1}, M) → Hom_{𝔫}(F_{q}, M) is zero, then the vertical differentials d_{v}^{p,q−1} of the double complex Hom_{𝔥}(P, Hom_{𝔫}(F, M)) are zero for all p. It now follows from the previous lemma that d_{r}^{p,q} and d_{r}^{p,q+r−2} are zero for all p and all r ≥ 2. □

Theorem 5.4. Let 0 → 𝔫 → 𝔤 → 𝔥 → 0 be the split extension determined by φ : 𝔥 → Der(𝔫) and let M be a 𝔤-module such that M^{𝔫} = M. Then the Hochschild-Serre spectral sequence associated to this extension with coefficients in M collapses at E_{2} in the following cases

(a) 𝔫 = 𝔫_{1} ⊕ 𝔫_{2}, where 𝔫_{1} is either abelian or free and 𝔫_{2} is either abelian or free;

(b) 𝔫 = 𝔫_{1} ∗𝔫_{2} ∗…∗𝔫_{k}, where each of the 𝔫_{i} is either abelian or free and φ(α)(𝔫_{i}) ⊆ 𝔫_{i} ∀α ∈ 𝔥 and i = 1, …, k.

Proof. First suppose that 𝔫 is abelian. We know that 𝔥 acts compatibly on the Chevalley-Eilenberg complex V (𝔫) of 𝔫. Since 𝔫 acts trivially on M, the differential

d^{q−1} : Hom_{𝔫}(V_{q−1}(𝔫), M) → Hom_{𝔫}(V_{q}(𝔫), M)

is zero for all q. Hence, it follows that d_{r}^{p,q} = 0 for all p, q and r ≥ 2 which means that the spectral sequence collapses at E_{2}.

Now, assume that 𝔫 = 𝔫_{1} ⊕𝔫_{2} with 𝔫_{1} and 𝔫_{2} both free. In Example 4.8, we constructed a resolution for 𝔫 that allows a compatible action in any case and one easily checks, by for example using the Künneth formula, that this resolution has zero differentials after applying Hom_{𝔫}(−, M), when 𝔫 acts trivially on M. So, just as before we obtain the desired collapse. If 𝔫 = 𝔫_{1} ⊕ 𝔫_{2}, with 𝔫_{1} free and 𝔫_{2} abelian, then Example 4.9 provides a free resolution of 𝔫 with compatible action that has zero differentials after applying Hom_{𝔫}(−, M) because 𝔫 acts trivially on M. Thus, the collapse follows and case (a) of the corollary is proven.

Part (b) is proven similarly by considering the compatible actions constructed in Lemma 4.5. □

Suppose that (8) is a split extension with a finite dimensional kernel and consider its associated Hochschild-Serre spectral sequence with coefficients in a 𝔤-module M,

E_{2}^{p,q} = H^{p}(𝔥, H^{q}(𝔫, M)) ⇒ H^{p+q}(𝔤, M).

It is clear that at some page t the Hochschild-Serre spectral sequence will collapse, i.e. E_{r} = E_{∞} for all r ≥ t. We define the length l of the spectral sequence to be the smallest t for which E_{t} = E_{∞}. This means that d_{r} = 0 for all r ≥ l, but d_{l−1} ≠ 0. Using the previous theorem we can now prove the following.

Theorem 5.5. Suppose 0 → 𝔫 → 𝔤 → 𝔥 → 0 is a split extension of Lie algebras such that dim_{k}(𝔫) = m < ∞ . Denote by (E_{r}, d_{r}) the associated Hochschild-Serre spectral sequence with coefficients in the 𝔤-module M. If 𝔫 acts trivially on M, then

(a) d_{r}^{p,m} = 0 for all p and all r ≥ 2;

(b) l ≤ max {2, m};

(c) H^{p}(𝔥, H^{m}(𝔫, M)) ⊕ H^{p+m}(𝔥, M) ⊆ H^{p+m}(𝔤, M) for all p.

Proof. Since 𝔫 acts trivially on M, either H^{m}(𝔫, M) = 0 or H^{m}(𝔫, M) ≅ M. If H^{m}(𝔫, M) = 0, then E_{r}^{p,m} = 0 for all p and all r ≥ 1. This of course implies d_{r}^{p,m} = 0 for all p and all r ≥ 2. If H^{m}(𝔫, M) = M, then d^{m−1} : Hom_{𝔫}((V_{m−1}(𝔫), M) → Hom_{𝔫}((V_{m}(𝔫), M) is zero. Since we always have a compatible action on the Chevalley-Eilenberg complex, Theorem 5.3 implies that d_{r}^{p,m} = 0 for all p and all r ≥ 2, so part (a) is proven.

Since 𝔫 acts trivially on M, we know that the differential d^{0} : Hom_{𝔫}((V_{0}(𝔫), M) → Hom_{𝔫}((V_{1}(𝔫), M) is zero. It follows that all differentials d_{r}, for r ≥ 2, that land on the bottom row of the spectral sequence are also zero. We conclude that l ≤ max {2, m}. This finishes (b).

A priori we have E_{∞}^{p,m} ⊕ E_{∞}^{p+m,0} ⊆ H^{p+m}(𝔤, M) and E_{∞}^{p+m,0} = H^{p+m}(𝔥, M) for all p. By part (a), E_{∞}^{p,m} = E_{m+1}^{p,m} = ⋯ = E_{2}^{p,m} for all p and E_{2}^{p,m} ≅ H^{p}(𝔥, H^{m}(𝔫, M)). This proves part (c). □

Remark 5.6.

- Since the extension splits and 𝔫 acts trivially on M, we know that the homomorphisms H^{p}(𝔥, M) → H^{p}(𝔤, M) are injective for every p. This is another way to see that all differentials d_{r}, for r ≥ 2, that land on the bottom row of the spectral sequence are zero.

- In [4], Barnes shows that the spectral sequence of split extensions of finite dimensional Lie algebra with abelian kernel collapses at the second page if the kernel acts trivially in the coefficients. Case (a) of Theorem 5.4 is a generalization of this result.

References

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[3] D. W. Barnes, Spectral sequence constructors in algebra in topology, Mem. Amer. Math. Soc. 53 No. 317, (1985)

[4] D. W. Barnes, On the length of the spectral sequence of a Lie algebra extension, Proceedings of the American Mathematical Society, 129, No. 2 (2001), 347–350

[5] T. Brady, Free resolutions for semi-direct products, Tohoku Math. J. (2) 45 No. 4 (1993), 535-537 [6] L. Evens, The Cohomology of Groups, Oxford Math. Monogr., Oxford Sci. Publ., The Clarendon

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[9] J. McClearly, A user’s guide to spectral sequences, Cambrigde University Press (2001)

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Department of Mathematics, Catholic University of Leuven, Kortrijk, Belgium

E-mail address: Dieter.Degrijse@kuleuven-kortrijk.be

Department of Mathematics, Catholic University of Leuven, Kortrijk, Belgium

E-mail address: Nansen.Petrosyan@kuleuven-kortrijk.be