Lie algebra cohomology

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 formula_1 is a compact simply connected Lie group, then it is determined by its Lie algebra, so it should be possible to calculate its cohomology from the Lie algebra. This can be done as follows. Its cohomology is the de Rham cohomology of the complex of differential forms on formula_1. Using an averaging process, this complex can be replaced by the complex of left-invariant differential forms. The left-invariant forms, meanwhile, are determined by their values at the identity, so that the space of left-invariant differential forms can be identified with the exterior algebra of the Lie algebra, with a suitable differential.
The construction of this differential on an exterior algebra makes sense for any Lie algebra, so it 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.
If formula_1 is a simply connected "noncompact" Lie group, the Lie algebra cohomology of the associated Lie algebra formula_4 does not necessarily reproduce the de Rham cohomology of formula_1. 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.
Let formula_4 be a Lie algebra over a commutative ring "R" with universal enveloping algebra formula_7, and let "M" be a representation of formula_4 (equivalently, a formula_7-module). Considering "R" as a trivial representation of formula_4, one defines the cohomology groups
(see Ext functor for the definition of Ext). Equivalently, these are the right derived functors of the left exact invariant submodule functor
Analogously, one can define Lie algebra homology as
(see Tor functor for the definition of Tor), which is equivalent to the left derived functors of the right exact coinvariants functor
Some important basic results about the cohomology of Lie algebras include Whitehead's lemmas, Weyl's theorem, and the Levi decomposition theorem.
Chevalley–Eilenberg complex.
Let formula_15 be a Lie algebra over a field formula_16, with a left action on the formula_15-module formula_18. The elements of the "Chevalley–Eilenberg complex"
are called cochains from formula_15 to formula_18. A homogeneous formula_22-cochain from formula_15 to formula_18 is thus an alternating formula_16-multilinear function formula_26. When formula_15 is finitely generated as vector space, the Chevalley–Eilenberg complex is canonically isomorphic to the tensor product formula_28, where formula_29denotes the dual vector space of formula_15.
The Lie bracket formula_31 on formula_15 induces a transpose application formula_33 by duality. The latter is sufficient to define a derivation formula_34 of the complex of cochains from formula_15 to formula_16 by extending formula_37according to the graded Leibniz rule. It follows from the Jacobi identity that formula_34 satisfies formula_39 and is in fact a differential. In this setting, formula_16 is viewed as a trivial formula_15-module while formula_42 may be thought of as constants.
In general, let formula_43 denote the left action of formula_15 on formula_18 and regard it as an application formula_46. The Chevalley–Eilenberg differential formula_47 is then the unique derivation extending formula_48 and formula_37 according to the graded Leibniz rule, the nilpotency condition formula_50 following from the Lie algebra homomorphism from formula_15 to formula_52 and the Jacobi identity in formula_15.
Explicitly, the differential of the formula_22-cochain formula_55 is the formula_56-cochain formula_57 given by:
formula_58
where the caret signifies omitting that argument.
When formula_1 is a real Lie group with Lie algebra formula_15, the Chevalley–Eilenberg complex may also be canonically identified with the space of left-invariant forms with values in formula_18, denoted by formula_62. The Chevalley–Eilenberg differential may then be thought of as a restriction of the covariant derivative on the trivial fiber bundle formula_63, equipped with the equivariant connection formula_64 associated with the left action formula_43 of formula_15 on formula_18. In the particular case where formula_68 is equipped with the trivial action of formula_15, the Chevalley–Eilenberg differential coincides with the restriction of the de Rham differential on formula_70 to the subspace of left-invariant differential forms.
Cohomology in small dimensions.
The zeroth cohomology group is (by definition) the invariants of the Lie algebra acting on the module:
The first cohomology group is the space of derivations modulo the space of inner derivations
where a derivation is a map formula_47 from the Lie algebra to formula_18 such that
and is called inner if it is given by 
for some formula_77 in formula_18.
The second cohomology group 
is the space of equivalence classes of Lie algebra extensions
of the Lie algebra by the module formula_18.
Similarly, any element of the cohomology group formula_82 gives an equivalence class of ways to extend the Lie algebra formula_15 to a "Lie formula_22-algebra" with formula_15 in grade zero and formula_18 in grade formula_22. A Lie formula_22-algebra is a homotopy Lie algebra with nonzero terms only in degrees 0 through formula_22.