User:Olivier Peltre/sandbox

In mathematics, Lie algebra cohomology is a cohomology theory for Lie algebras. It was first introduced in 1929 by Élie Cartan to study the topology of Lie groups and homogeneous spaces by relating cohomological methods of Georges de Rham to properties of the Lie algebra. It was later extended by to coefficients in an arbitrary Lie module.

Motivation
If $$G$$ is a compact simply connected Lie group, it is determined by its Lie algebra, so it should be possible to access its de Rham cohomology cohomology from the Lie algebra. When $$G$$ is a compact Lie group, the de Rham complex of differential forms on $$G$$ may projected onto the complex of left-invariant differential forms using an averaging process. Left-invariant forms are determined by their values at the identity, so that the space of left-invariant differential forms may be identified with the exterior algebra of the dual vector space of the Lie algebra and inherit from the differential of the de Rham complex.

The construction of this differential on an exterior algebra makes sense for any Lie algebra, so is used to define Lie algebra cohomology for all Lie algebras. More generally one uses a similar construction to define Lie algebra cohomology with coefficients in a module.

It should be noted that if $$G$$ is a simply connected noncompact Lie group, the Lie algebra cohomology of the associated Lie algebra $$\mathfrak g$$ does not necessarily reproduce the de Rham cohomology of $$G$$. The reason for this is that the passage from the complex of all differential forms to the complex of left-invariant differential forms uses an averaging process that only makes sense for compact groups.

Definition of the Chevalley–Eilenberg complex
Let $$\mathfrak{g}$$ be a Lie algebra over a field $$k$$, with a left action on the $$\mathfrak{g}$$-module $$M$$. The elements of the Chevalley–Eilenberg complex
 * $$\mathrm{Hom}_k(\Lambda^\cdot\mathfrak{g},M) \simeq M \otimes \Lambda^\cdot \mathfrak{g}^*$$

are called cochains of $$\mathfrak{g}$$ in $$M$$. A homogeneous $$n$$-cochain of $$\mathfrak{g}$$ in $$M$$ is thus an alternating $$k$$-multilinear function $$f\colon\Lambda^n\mathfrak{g}\to M$$. They are simply called cochains of $$\mathfrak{g}$$ when $$M=k$$ is the trivial $$\mathfrak{g}$$-module.

Scalar coefficients
The Lie bracket $$C: \Lambda^2 \mathfrak{g} \rightarrow \mathfrak{g}$$ on $$\mathfrak{g}$$ induces a transpose application $$C^* \colon \mathfrak{g}^* \rightarrow \Lambda^2 \mathfrak{g}^*$$ by duality. Extending $$d_1 = -C^*$$ according to the graded Leibniz rule defines a unique derivation $$d$$ of the complex of cochains of $$\mathfrak{g}$$ in $$k$$. It follows from the Jacobi identity that $$d$$ satisfies $$d^2 = 0$$ and is actually a differential.

Explicitly, one has for every $$\xi \in \mathfrak{g}^*$$ and every $$x,x' \in \mathfrak{g}$$:  "$d \xi(x,x') = - \xi([x,x'])$"and for every $$\alpha, \beta \in \Lambda^{\cdot}\mathfrak{g}^*$$:"\alpha"denoting by $$|\alpha|$$ the degree of $$\alpha$$.

Coefficients in a module
More generally, let $$\omega \in \mathrm{Hom(\mathfrak{g}, \mathrm{End}(M))}$$ denote the left action of $$\mathfrak{g}$$ on $$M$$ and regard it as an application $$d_0 \colon M \rightarrow M \otimes \mathfrak{g}^*$$. The Chevalley–Eilenberg differential $$d$$ is then the unique derivation extending $$d_0 \otimes 1 + 1 \otimes d_1$$according to the graded Leibniz rule, the nilpotency condition $$d^2 = 0$$ following from the Lie algebra homomorphism from $$\mathfrak{g}$$ to $$\mathrm{End}(M)$$ and the Jacobi identity in $$\mathfrak{g}$$.

Explicitly, one has for every $$m \in M$$:  "$dm(x) = x \cdot m$"and for every $$m \alpha \in M \otimes \Lambda^\cdot \mathfrak{g}^*$$:"$d(m \alpha) = dm\wedge \alpha + m d\alpha$"

General formula
The differential of the $$n$$-cochain $$\alpha$$ is the $$(n+1)$$-cochain $$d\alpha$$ given by :

$$\begin{align} d \alpha \left(x_1, \ldots, x_{n+1}\right) = &\sum_i    (-1)^{i+1}\, x_i\cdot \alpha\left(x_1, \ldots, \hat x_i, \ldots, x_{n+1}\right) \\ + &\sum_{i<j}\, (-1)^{i+j}\,  \alpha\left(\left[x_i, x_j\right], x_1, \ldots, \hat x_i, \ldots, \hat x_j, \ldots, x_{n+1}\right) \end{align}$$

where the caret signifies omitting that argument.

Relation with Lie group actions
When $$G$$ is a real Lie group with Lie algebra $$\mathfrak{g}$$, the Chevalley–Eilenberg may also be canonically identified with the space of left-invariant forms with values in $$M$$ denoted by $$\Omega^{\bullet}(G,M)^G$$. In the particular case where $$M = k = \mathbb{R}$$ is equipped with the trivial action of $$\mathfrak{g}$$, the Chevalley–Eilenberg differential coincides with the restriction of the de Rham differential on $$\Omega^{\bullet}(G)$$ to the subspace of left-invariant differential forms.

In general, the Chevalley–Eilenberg differential may be thought of as a restriction of the covariant derivative on the trivial fiber bundle $$G \times M \rightarrow G$$, equipped with the left-invariant connection $$\tilde{\omega} \in \Omega^1(G,\mathrm{End}(M))$$ induced by the infinitesimal left action $$\omega$$ of $$\mathfrak{g}$$ on $$M$$.

The fact that $$\omega$$ is a Lie algebra homomorphism implies that $$\tilde{\omega}$$ is right-equivariant and defines a flat connection on $$G \times M$$. Equivalently, the graph of $$\tilde{\omega}$$ defines a Lie algebra homomorphism from the space $$\Alpha^1(G)$$ of tangent vector fields on $$G$$ to the space $$\Alpha^1(G \times M)$$ of tangent vector fields on $$G \times M$$. The image of this homomorphism is a locally integrable distribution of tangent subspaces in $$T(G \times M)$$ and the leaves of this foliation locally integrate the infinitesimal action of $$\mathfrak{g}$$ on $$M$$.

When $$G$$ is simply connected, these leaves are the graph of a Lie group action of $$G$$ on $$M$$, their projections on $$M$$ being the orbits under $$G$$. More precisely, the leaf passing through the point $$(1,m)\in G \times M$$ cuts the fiber of $$g \in G$$ at $$(g,g\cdot m)$$ while its projection in $$M$$ is the orbit $$G\cdot m \subseteq M$$.

Lift of the action
When $$\omega \in \mathrm{Hom}(\mathfrak{g}, \mathrm{End}(M))$$ is an action of $$\mathfrak{g}$$ on $$M$$, it also induces an action of $$\mathfrak{g}$$ on the $$\mathfrak{g}$$-module $$\mathrm{End}(M)$$.

Then $$\omega$$ is a $$1$$-cocycle of $$\mathfrak{g}$$ in $$\mathrm{End}(M)$$, as for every $$x,x' \in \mathfrak{g}$$: "$d\omega(x,x') = [\omega(x),\omega(x')] - \omega([x,x']) = 0$"expresses that $$\omega$$ is a Lie algebra morphism.

It is a also a $$1$$-coboundary, as $$\omega = d(\mathrm{id_M})$$ so that $$\omega$$ has a null cohomology class in $$H(\mathfrak{g},\mathrm{End}(M))$$.

Adjoint representation
Considering $$\mathfrak{g}$$ as a $$\mathfrak{g}$$-module through its adjoint representation, we have for every $$0$$-cochain $$x \in \mathfrak{g}$$: "$dx = \mathrm{ad}_x$" so that a $$0$$-cocycle in $$\mathfrak{g}$$ is an element of the center of $$\mathfrak{g}$$: "$H^0(\mathfrak{g},\mathfrak{g}) = Z(\mathfrak{g})$"

A $$1$$-cochain $$\alpha \in \mathfrak{g} \otimes \mathfrak{g}^*$$ is a linear endomorphism of $$\mathfrak{g}$$ and for every $$x,x' \in \mathfrak{g}$$: "$d\alpha(x,x') = [\alpha(x),x'] + [x,\alpha(x')] - \alpha([x,x'])$" so that a $$1$$-cocycle in $$\mathfrak{g}$$ is a derivation of $$\mathfrak{g}$$ and: "$H^1(\mathfrak{g},\mathfrak{g}) = \mathrm{Der}(\mathfrak{g})/\mathrm{ad}_\mathfrak{g}$" is the space of derivations of $$\mathfrak{g}$$ modulo the inner derivations of $$\mathfrak{g}$$.

A $$2$$-cochain $$\beta \in \mathfrak{g} \otimes \Lambda^2 \mathfrak{g}^*$$ is an alternating bilinear map from $$\mathfrak{g}\times\mathfrak{g} $$ to $$\mathfrak{g}$$, and for every $$x_1,x_2,x_3 \in \mathfrak{g}$$:"$d\beta(x_1,x_2,x_3) = \sum_{(ijk)\in\mathfrak{A}_3} [x_i,\beta(x_j,x_k)] + \beta(x_i,[x_j,x_k]) $"where $$\mathfrak{A}_3$$ denotes the three circular permutations of $$\{1,2,3\}$$.

Associating to $$\beta$$ a deformed bracket $$[\cdot,\cdot]_{\beta}$$ defined by:"$[x,x']_{\beta} = [x,x'] + \beta(x,x')$"a $$2$$-cocycle $$\beta$$ in $$\mathfrak{g}$$ may be interpreted as an infinitesimal deformation $$[\cdot,\cdot]_{\beta}$$ of the original Lie bracket satisfying the Jacobi identity up to quadratic terms in $$\beta$$.

Category-theoretical definition
Let $$\mathfrak g$$ be a Lie algebra over a commutative ring R with universal enveloping algebra $$U\mathfrak g$$, and let M be a representation of $$\mathfrak g$$ (equivalently, a $$U\mathfrak g$$-module). Considering R as a trivial representation of $$\mathfrak g$$, one defines the cohomology groups


 * $$\mathrm{H}^n(\mathfrak{g}; M) := \mathrm{Ext}^n_{U\mathfrak{g}}(R, M)$$

(see Ext functor for the definition of Ext). Equivalently, these are the right derived functors of the left exact invariant submodule functor


 * $$M \mapsto M^{\mathfrak{g}} := \{ m \in M \mid xm = 0\ \text{ for all } x \in \mathfrak{g}\}.$$

Analogously, one can define Lie algebra homology as


 * $$\mathrm{H}_n(\mathfrak{g}; M) := \mathrm{Tor}_n^{U\mathfrak{g}}(R, M)$$

(see Tor functor for the definition of Tor), which is equivalent to the left derived functors of the right exact coinvariants functor


 * $$ M \mapsto M_{\mathfrak{g}} := M / \mathfrak{g} M.$$

Some important basic results about the cohomology of Lie algebras include Whitehead's lemmas, Weyl's theorem, and the Levi decomposition theorem.

Cohomology in small dimensions
The zeroth cohomology group is (by definition) the invariants of the Lie algebra acting on the module:
 * $$H^0(\mathfrak{g}; M) =M^{\mathfrak{g}} = \{ m \in M \mid xm = 0\ \text{ for all } x \in \mathfrak{g}\}.$$

The first cohomology group is the space $Der$ of derivations modulo the space $Ider$ of inner derivations
 * $$H^1(\mathfrak{g}; M) = \mathrm{Der}(\mathfrak{g}, M)/\mathrm{Ider} (\mathfrak{g}, M)$$

where a derivation is a map $$d$$ from the Lie algebra to $$M$$ such that
 * $$d[x,y] = xdy-ydx~$$

and is called inner if it is given by
 * $$dx = xa~$$

for some $$a$$ in $$M$$.

The second cohomology group
 * $$H^2(\mathfrak{g}; M)$$

is the space of equivalence classes of Lie algebra extensions
 * $$0\rightarrow M\rightarrow \mathfrak{h}\rightarrow\mathfrak{g}\rightarrow 0$$

of the Lie algebra by the module $$M$$.

There do not seem to be any similar easy interpretations for the higher cohomology groups.