Lie Algebra Cohomology — Mathematics Knowledge System

Overview

Lie algebra cohomology is the cohomology theory for Lie algebras and their modules, developed by Chevalley and Eilenberg in 1948. For a Lie algebra g and a g-module M, the cohomology H^n(g, M) is defined via the Chevalley-Eilenberg cochain complex. Key results include: H^0(g,M) = M^g (invariants), H^1(g,M) classifies derivations modulo inner derivations, H^2(g,M) classifies extensions. Whitehead's lemmas show that for semisimple g, H^1(g,M)=H^2(g,M)=0 (for finite-dimensional M), implying complete reducibility (Weyl's theorem).

Intuition

Lie algebra cohomology measures 'obstructions' and 'invariants' in the theory of Lie algebra representations. H^1(g, g) = 0 for semisimple g says there are no outer derivations (all derivations are inner -- the algebra is 'rigid'). H^2(g, M) = 0 says every extension of g-modules splits -- this is the key to Weyl's complete reducibility theorem. The Chevalley-Eilenberg complex looks like the de Rham complex on the Lie group, but works purely algebraically.

Formal Definition

Definition

For a Lie algebra g over k and a g-module M, the Chevalley-Eilenberg cochain complex is: C^n(g, M) = Hom_k(wedge^n g, M) with differential d: C^n -> C^{n+1} given by (df)(x_0,...,x_n) = sum_{i<j} (-1)^{i+j} f([x_i,x_j], x_0,...,hat{x_i},...,hat{x_j},...,x_n) + sum_i (-1)^i x_i * f(x_0,...,hat{x_i},...,x_n). The Lie algebra cohomology is H^n(g,M) = ker(d^n)/im(d^{n-1}).

C^n(\mathfrak{g}, M) = \operatorname{Hom}_k(\wedge^n \mathfrak{g},\, M)

Chevalley-Eilenberg cochains

(df)(x_0,\ldots,x_n) = \sum_{i<j}(-1)^{i+j}f([x_i,x_j],\ldots) + \sum_i(-1)^i x_i\cdot f(\ldots)

Chevalley-Eilenberg differential

H^n(\mathfrak{g}, M) = \ker(d^n)/\operatorname{im}(d^{n-1})

Lie algebra cohomology

H^0(\mathfrak{g},M) = M^{\mathfrak{g}} = \{m \in M : x\cdot m = 0\;\forall x\in\mathfrak{g}\}

H^0 = invariants

Notation

Notation	Meaning
$C^n(\mathfrak{g},M)$	Degree-n Chevalley-Eilenberg cochains
$H^n(\mathfrak{g},M)$	Degree-n Lie algebra cohomology
$M^{\mathfrak{g}}$	g-invariants in M
$\wedge^n\mathfrak{g}$	n-th exterior power of g

Theorems

Theorem 1: Whitehead's First Lemma

Let g be a semisimple Lie algebra over a field of characteristic 0 and M a finite-dimensional g-module. Then H^1(g, M) = 0. Equivalently, every derivation of g with values in M is inner (of the form m -> [x, m] for some fixed m in M when M=g), and every 1-cocycle is a coboundary.

Theorem 2: Whitehead's Second Lemma

Under the same hypotheses, H^2(g, M) = 0. This implies that every short exact sequence of g-modules 0 -> M -> E -> N -> 0 splits (E is isomorphic to M direct-sum N as g-modules). As a corollary: every finite-dimensional representation of a semisimple Lie algebra over characteristic 0 is completely reducible (Weyl's theorem).

Theorem 3: Hochschild-Serre Spectral Sequence

If h is an ideal in g, there is a spectral sequence E_2^{p,q} = H^p(g/h, H^q(h, M)) converging to H^{p+q}(g, M). This allows computing cohomology of g from the cohomology of h and the quotient g/h.

Worked Examples

1
C^0(g,k) = k, C^1(g,k) = Hom(g, k) = g^* (dual), C^2(g,k) = Hom(wedge^2 g, k).
2
d: C^0 -> C^1: (d c)(x) = x*c = 0 (trivial action). So im(d^0) = 0.
3
d: C^1 -> C^2: (d f)(x,y) = f([x,y]) - x*f(y) + y*f(x) = f([x,y]) (trivial action). So ker(d^1) = {f : f([x,y])=0 for all x,y} = (g/[g,g])^*.
$\ker(d^1) = (\mathfrak{g}/[\mathfrak{g},\mathfrak{g}])^*$
4
H^1(g, k) = ker(d^1)/im(d^0) = (g/[g,g])^*. This is the dual of the abelianization.
$H^1(\mathfrak{g}, k) \cong (\mathfrak{g}/[\mathfrak{g},\mathfrak{g}])^*$

✓ Answer

H^1(g, k) = dual of the abelianization g/[g,g]. For semisimple g, [g,g]=g so H^1(g,k)=0.

Practice Problems

Hardfree response

Explain how H^2(g, M) = 0 implies that every finite-dimensional g-module is completely reducible.

Common Mistakes

Common Mistake

Confusing Lie algebra cohomology with group cohomology.

For a connected simply connected Lie group G with Lie algebra g, H^n(g, R) agrees with the de Rham cohomology H^n(G, R) as vector spaces (van Est theorem). But for discrete groups, group cohomology H^n(G, M) uses a different cochain complex (bar resolution). The two theories agree only for connected algebraic groups in characteristic zero via passage through the Lie algebra.

Quiz

H^2(g, M) = 0 for semisimple g and finite-dimensional M implies:

g has no non-trivial modulesEvery extension of g-modules splits (complete reducibility)g is abelianEvery derivation of g is zero

Historical Background

Elie Cartan had computed the cohomology of Lie groups via differential forms. Chevalley and Eilenberg (1948) gave a purely algebraic treatment defining the cochain complex now bearing their name. This allowed Lie algebra cohomology to be studied without reference to the underlying manifold. Koszul (1950) connected this to the Ext functor in homological algebra. Hochschild and Serre established the spectral sequence relating Lie algebra, subalgebra, and quotient cohomology.

1929
Cartan uses differential forms to compute topology of Lie groups
Elie Cartan
1948
Chevalley-Eilenberg define the cochain complex for Lie algebra cohomology
Claude Chevalley, Samuel Eilenberg
1953
Hochschild-Serre spectral sequence for Lie algebra cohomology
Gerhard Hochschild, Jean-Pierre Serre
1970s
Connections to BRST cohomology in physics established
Carlo Becchi, Alain Rouet, Raymond Stora, Igor Tyutin

Summary

H^n(g, M) is defined via the Chevalley-Eilenberg complex Hom(wedge^n g, M) with an explicit differential.
H^0(g,M) = g-invariants, H^1 = derivations mod inner, H^2 = extensions (Ext^1).
Whitehead's lemmas: for semisimple g and finite-dimensional M over char 0, H^1=H^2=0.
Consequence: complete reducibility (Weyl's theorem) follows from H^2=0.

References

BookWeibel, C. An Introduction to Homological Algebra. Cambridge, 1994.
BookKnapp, A.W. Lie Groups Beyond an Introduction. Birkhauser, 2002.
WebsiteWikipedia -- Lie algebra cohomology