4. LIE ALGEBRA COHOMOLOGY

This is a quick-and-dirty introduction to Lie algebra cohomology. A reasonable setting in which to work is this: k is a base field (any characteristic is OK, and algebraic closure doesn’t matter). We fix a Lie algebra 𝔤 over k, a subalgebra 𝔮 of 𝔤, and an ideal 𝔲 of 𝔮:

𝔤 ⊃ 𝔮 ⊃ 𝔲,  [𝔮, 𝔲] ⊂ 𝔲.  (4.1)

The main example we’ll be interested in is 𝔮 a parabolic subalgebra of a reductive 𝔤, with nil radical 𝔲; but not even finite-dimensionality of the algebras matters for the definitions.

The definition of Lie algebra cohomology lives in the world of modules over rings. We write

M(𝔤) = category of modules (over k) for the Lie algebra 𝔤.  (4.2)

This is the same thing as the category of modules over the associative ring U(𝔤), so we can use freely ideas from that subject. Our point of view is largely that of [CE]: we will manage with nothing more abstract than categories of modules, and derived categories will remain only a distant rumor. Complete details and much more can be found also in [Knapp].

A covariant functor F from a category A to another category B associates to each object M of A an object F(M) of B; and to each map f from objects M to N of A, associates a map F(f) from F(M) to F(N). These associations are required to send the identity map to the identity map, and to respect composition. A contravariant functor G is formally similar except that it reverses all arrows, associating to each map g from M to N a map G(g) from F(N) to F(M).

We can now say that 𝔲-cohomology is a collection of functors H^{i}(𝔲, ·), for i = 0, 1, 2, …, each going from M(𝔤) to M(𝔮/𝔲). Before defining these functors, I will state three of their fundamental properties. (One natural perspective is that these properties are the definition; the general arguments of homological algebra show how to construct uniquely functors with these properties. But I will write a more mundane definition.)

H^{0}(𝔲, M) = {m ∈ M | x · m = 0, all x ∈ 𝔲} =_{def} M^{𝔲}.  (4.3)(a)

Whenever

0 → A → B → C → 0  (4.3)(b)

is a short exact sequence of 𝔤-modules, there is long exact sequence of 𝔮/𝔲-modules

\begin{aligned}0 & → & H^{0}(𝔲, A) & → & H^{0}(𝔲, B) & → & H^{0}(𝔲, C) \\ & → & H^{1}(𝔲, A) & → & H^{1}(𝔲, B) & → & H^{1}(𝔲, C) \\ & → & H^{2}(𝔲, A) & → & H^{2}(𝔲, B) & → & H^{2}(𝔲, C) & ⋯ \end{aligned}  (4.3)(c)

Typeset by AMS-TEX

Here the maps within each row (like H^{i}(𝔲, B) → H^{i}(𝔲, C)) arise by applying the functor H^{i} to the maps in (4.3)(b). The maps from one row to the next (like H^{i}(𝔲, C) → H^{i+1}(𝔲, A)) are called connecting homomorphisms, and are something new.

To state the third fundamental property of cohomology, we need a notion from module theory. A (left) module I for a ring R is called injective if the contravariant functor Hom_{R}(·, I) (from R-modules to abelian groups) is exact. Making this explicit in our case, the requirement on I is that if M is any 𝔤-module and S is any submodule, then any 𝔤-module homomorphism from S to I can be extended to a 𝔤-module homomorphism from M to I. Pictorially, the assertion is that if we are given maps corresponding to the first two arrows in the following diagram, then there exists a map corresponding to the last (vertical) arrow so that the diagram commutes.

\begin{aligned}S & ↪ & M \\ & ↘ & ↓ \\ & & I \end{aligned}  (I)

The notion of injective module is formally very similar to the (possibly more familiar) notion of projective module. An R-module P is called projective if the covariant functor Hom_{R}(P, ·) from R-modules to abelian groups is exact. This means that if N is any R-module and Q any quotient of N, then any R-module homomorphism from P to Q lifts to a homomorphism from P to N. Here is the corresponding picture; the assertion is that if we are given the two arrows on the right, then we can find a vertical map on the left making the diagram commute.

\begin{matrix}N & ↠ & Q \\ ↑ & ↗ \\ P \end{matrix}  (P)

The third fundamental property is that whenever I is an injective 𝔤-module, then

H^{i}(𝔲, I) = 0,  all i > 0.  (4.3)(d)

I’ll turn next to a direct and mundane definition of Lie algebra cohomology. One almost trivial bit of linear algebra is helpful.

Lemma 4.4. Suppose V and W are vector spaces over a field k. Write ∧^{i}V for the ith exterior power of V, the quotient of the ith tensor power by the relation making multiplication antisymmetric. Then there is a natural identification

Hom_{k}(∧^{i}V, W) ≃ \{ω: \underbrace{V × ⋯ × V}_{i factors} → W | ω is linear in each variable and

vanishes if two arguments are equal}.

Definition 4.5. In the setting (4.1), suppose M is a 𝔤-module. Regard

Hom_{k}(∧^{i}𝔲, M)  (4.5)(a)

as a 𝔮-module, using Hom of the (ith exterior power of the) adjoint action of 𝔮 on 𝔲 into the (restriction of the action of 𝔤 to the) action of 𝔮 on M. Define

d: Hom_{k}(∧^{i}𝔲, M) → Hom_{k}(∧^{i+1}𝔲, M),  (4.5)(b)

dω(X_{0} … X_{i}) = ∑_{j=0}^{i}(−1)^{j}X_{j} · ω(X_{0} … X̂_{j} … X_{i})
+ ∑_{0≤j<k≤i}(−1)^{j+k}ω([X_{j}, X_{k}], X_{0} … X̂_{j} … X̂_{k} … X_{i}).(4.5)(c)

Here a hat over an argument means that it is omitted. We use the identification in Lemma 4.4 throughout. An elementary calculation shows that d intertwines the action of 𝔮, and that d^{2} = 0. The maps d therefore make the spaces (4.5)(a) into a chain complex of 𝔮-modules. We define

H^{i}(𝔲, M) = ith cohomology of the complex
= ker d on Hom_{k}(∧^{i}𝔲, M)/im d on Hom_{k}(∧^{i−1}𝔲, M) (4.5)(d)

As I said in class, the formula for d mimics a standard one in differential geometry. There ω is an i-form on a manifold, regarded as a (special kind of) i-linear map from vector fields to smooth functions. At this point one should pause to verify that the functors H^{i} of Definition 4.5 are really functors, and that they satisfy the requirements (4.3). If you accept the assertion in the definition that d^{2} is easily seen to be zero, and that d respects the action of 𝔮, what follows is only that H^{i}(𝔲, M) is a 𝔮-module. We need to see that 𝔲 acts by zero on the cohomology. For that, suppose X ∈ 𝔲. Interior multiplication by X is a map

ι(X): Hom_{k}(∧^{i}𝔲, M) → Hom_{k}(∧^{i−1}𝔲, M),  (4.6)(a)
(ι(X)ω)(X_{2} … X_{i}) = ω(X, X_{2} … X_{i}).  (4.6)(b)

Write θ(X) for the action of X on the complex of Definition 4.5. Then another straightforward calculation shows that

dι(X) + ι(X)d = θ(X),  (4.6)(c)

both sides being linear maps from Hom_{k}(∧^{i}𝔲, M) to itself. (This is again an analogue of a formula from differential geometry.) If now ω is an i-cocycle (an element of the kernel of d), then it follows that

X · ω = θ(X)ω = d(ι(X)ω),  (4.6)(d)

so that X · ω is a coboundary. (In differential geometry, the conclusion is that the action of vector fields on differential forms by Lie derivative is zero on de Rham cohomology.) This shows that H^{i}(𝔲, M) is 𝔮/𝔲-module.

The complex defining the cohomology is functorial: any 𝔤-module map from M to M′ defines a 𝔮-module map of complexes

Hom_{k}(∧^{∗}𝔲, M) → Hom_{k}(∧^{∗}𝔲, M′),

which induces 𝔮-module maps on cohomology. In the setting of (4.3)(b), the sequence of complexes

0 → Hom_{k}(∧^{∗}𝔲, A) → Hom_{k}(∧^{∗}𝔲, B) → Hom_{k}(∧^{∗}𝔲, C) → 0

is exact, and the existence of connecting homomorphisms as required in (4.3)(c) follows by the usual diagram chase.

The property (4.3)(d) (that higher cohomology vanishes for injective 𝔤-modules) requires a little more work; and you may wonder why anyone one should bother. For pedagogical reasons, I’ll outline the work first; but you may wish to skip ahead to the Casselman-Osborne theorem, to see a reason to bother.

In order to verify (4.3)(d), we need to understand something about what injective 𝔤-modules look like. The theory for projective modules is formally parallel but perhaps a little more familiar, so I will discuss that at the same time. The general results about rings and modules here are from [CE], II.6.

Definition 4.7. Suppose R and S are associative rings with unit, and ϕ: S → R is a ring homomorphism. If M is any (left) S-module, then

ϕM =_{def} R ⊗_{S} M  (4.7)(a)

is a (left) R-module. This defines a covariant functor from S-modules to R-modules, called extension of scalars. This functor is a left adjoint to the “forgetful” functor F making any R-module into an S-module. This statement means that if N is any left R-module, then there is a natural isomorphism (the “adjunction formula”)

Hom_{R}(_{ϕ}M, N) ≃ Hom_{S}(M, FN).  (4.7)(b)

The isomorphism identifies a map t on the right with T on the left if

T (r ⊗ m) = r · t(m),  t(m) = T (1 ⊗ m).  (4.7)(c)

The functor F is obviously exact, because it doesn’t change N as a set. The exactness of F and the adjunction formula together imply that extension of scalars is right exact. If R is flat as a right S-module—for example, if it is free—then the tensor product formula shows that extension of scalars is exact.

Similarly, we can define a left R-module

ϕM =_{def} Hom_{S}(R, M)  (4.7)(d)

Here the action of R on the right is given by

(r · μ)(x) = μ(xr)  (μ ∈ ^{ϕ}M, r ∈ R, x ∈ R).  (4.7)(e)

This is also a covariant functor from S-modules to R-modules, which we will call coextension of scalars. (There doesn’t seem to be a good name for it. Cartan and Eilenberg say “contravariant extension of scalars,” which seems terrible because it’s a covariant functor.) This functor is a right adjoint to the forgetful functor: if N is any left R-module, then there is a natural isomorphism (the “adjunction formula”)

Hom_{R}(N,^{ϕ}M) ≃ Hom_{S}(FN, M).  (4.7)(f)

This time the isomorphism identifies a map T on the left with t on the right if

T (n)(r) = r · t(n),  t(n) = T (n)(1).  (4.7)(g)

The exactness of F and the adjunction formula together imply that coextension of scalars is left exact. If R is projective as a left S-module—for example, if it is free—then the definition of ^{ϕ}M as a Hom shows that coextension of scalars is exact.

Proposition 4.8. ([CE], Proposition II.6.1) Suppose we are in the setting of Definition 4.7.

(1) If P is a projective S-module, then the extension of scalars

ϕP = R ⊗_{S} P

is a projective R-module.

(2) Suppose that M is any R-module, that P is an S-module, and that q: P → FM is a surjective map of S-modules. Then the R-module homomorphism

Q:_{ϕ}P → M

corresponding to q under the adjunction isomorphism (4.7)(b) is a surjective map of R-modules.

(3) If I is an injective S-module, then the coextension of scalars

ϕI = Hom_{S}(R, I)

is an injective R-module.

(4) Suppose that M is any R-module, that I is an S-module, and that j: FM → I is an injective map of S-modules. Then the R-module homomorphism

J: M → ^{ϕ}I

corresponding to j under the adjunction isomorphism (4.7)(f) is an injective map of R-modules.

Proof. For (1), according to the adjunction formula (4.7)(b),

Hom_{R}(_{ϕ}P, N) ≃ Hom_{S}(P, FN).

The right side is exact in N, because the forgetful functor is exact and P is projective. The proof for (3) is identical. For (2), the formula (4.7)(c) shows that Q is surjective (already on the subset of elements 1 ⊗ p) if q is. The proof for (4) is identical. □

One of the basic facts of homological algebra is that any R-module is a quotient of a projective R-module, and a submodule of an injective R-module. Proposition 4.8 reduces the proof of this fact to the case R = ℤ; a proof in that case can be found in [CE], VII.5. The rings we are interested in all contain fields; and in that case Proposition 4.8 reduces the fact to the case of fields. Any module over a field (that is to say, any vector space) is both projective and injective (assuming the axiom of choice); for injectivity, this is just the statement that any linear map defined on a subspace can be extended linearly to the whole vector space. Here is a statement in the case of enveloping algebras.

Corollary 4.9. Suppose that 𝔤 is a Lie algebra over a field k.

(1) Suppose V is a vector space over k. Then Hom_{k}(U(𝔤), V) is an injective 𝔤-module.

(2) Any 𝔤-module is a submodule of some Hom_{k}(U(𝔤), V).

(3) Any injective 𝔤-module is a direct summand of some Hom_{k}(U(𝔤), V).

Proof. Since V is an injective k-module, (1) and (2) follow from Proposition 4.8. For (3), given an injective 𝔤-module I, construct an inclusion I ↪ Hom_{k}(U(𝔤), V), and write I′ for the image of I. According to the injectivity of I, the 𝔤-module isomorphism I′ → I (the inverse of the inclusion) extends to a 𝔤-module map

ξ: Hom_{k}(U(𝔤), V) → I.

It’s easy to see that Hom_{k}(U(𝔤), V) is the direct sum of I′ and the kernel of ξ. □

Here is the proof of property (4.3)(d).

Theorem 4.10. In the setting (4.1), suppose I is an injective U(𝔤)-module. Then H^{i}(𝔲, I) = 0 for i > 0.

Sketch of proof. By Corollary 4.9, (and since the cohomology of a direct sum is obviously the direct sum of the cohomologies) we may assume that I = Hom_{k}(U(𝔤), V). By the Poincaré-Birkhoff-Witt Theorem, U(𝔤) is free as a right U(𝔲)-module. We can therefore find a vector subspace W of U(𝔤) so that U(𝔤) ≃ W ⊗_{k} U(𝔲). It follows that

I ≃ Hom_{k}(U(𝔲), Hom_{k}(W, V)) ≃ Hom_{k}(U(𝔲), H)

as a U(𝔲)-module; here H is the vector space Hom_{k}(W, V).

Now consider the complex computing Lie algebra cohomology of I. The terms are

Hom_{k}(∧^{i}𝔲, Hom_{k}(U(𝔲), H)) ≃ Hom_{k}(U(𝔲) ⊗ ∧^{i}𝔲, H).

The differentials arise as Hom(·, H) of boundary operators

∂: U(𝔲) ⊗ ∧^{i+1}𝔲 → U(𝔲) ⊗ ∧^{i}𝔲,  (4.11)(a)

∂(u ⊗ X_{0} ∧ ⋯ ∧ X_{i}) = ∑_{j}(−1)^{j}uX_{j} ⊗ X_{0} ∧ ⋯ X̂_{j} ⋯ ∧ X_{i}
+ ∑_{j<k}(−1)^{j+k}u ⊗ [X_{j}, X_{k}] ∧ ⋯ X̂_{j} ⋯ X̂_{k} ⋯ ∧ X_{i}. (4.11)(b)

It is therefore sufficient to prove that the homology of this last complex vanishes except in degree 0.

For that, recall the standard increasing filtration of U(𝔲): the subspace U_{n}(𝔲) is the span of products of at most n terms in 𝔲. If we define

C_{n}^{i} = U_{n−i}(𝔲) ⊗ ∧^{i}𝔲,  (4.11)(c)

then it is clear from (4.11)(b) that

C_{0}^{i} ⊂ C_{1}^{i} ⊂ ⋯,  ⋃_{n} C_{n}^{i} = U(𝔲) ⊗ ∧^{i}𝔲,  ∂: C_{n}^{i+1} → C_{n}^{i}.  (4.11)(d)

The Poincaré-Birkhoff-Witt Theorem allows us to identify

C_{n}^{i}/C_{n−1}^{i} ≃ S^{n−i}(𝔲) ⊗ ∧^{i}𝔲.  (4.11)(e)

According to (4.11)(d), the boundary operator defines by passage to the quotient

∂_{n}: S^{n−(i+1)}(𝔲) ⊗ ∧^{i+1}𝔲 → S^{n−i}(𝔲) ⊗ ∧^{i}𝔲.  (4.11)(f)

This is a Koszul complex. It is a standard and elementary fact that this complex is exact except at i = n = 0 (where the homology is one-dimensional). I have not found a really convenient reference for this statement; it is the content of IV.6 in [Knapp].

Another elementary argument (also in IV.6 of [Knapp]) passes from the exactness of the associated graded complex back to exactness of (4.11)(a), and proves the theorem. □

If I get ambitious I’ll add an account of the Casselman-Osborne theorem.

References

[CE] H. Cartan and S. Eilenberg, Homological Algebra, Princeton University Press, Princeton, New Jersey, 1956.

[Knapp] A. Knapp, Lie Groups, Lie Algebras, and Cohomology, Mathematical Notes 34, Princeton University Press, Princeton, New Jersey, 1988.