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 Claude Chevalley and Samuel Eilenberg (1948) to coefficients in an arbitrary Lie module.
Motivation If G {\displaystyle G} 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 G {\displaystyle G} . 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 G {\displaystyle G} is a simply connected noncompact Lie group, the Lie algebra cohomology of the associated Lie algebra g {\displaystyle {\mathfrak {g}}} does not necessarily reproduce the de Rham cohomology of G {\displaystyle 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 Let g {\displaystyle {\mathfrak {g}}} be a Lie algebra over a commutative ring R with universal enveloping algebra U g {\displaystyle U{\mathfrak {g}}} , and let M be a representation of g {\displaystyle {\mathfrak {g}}} (equivalently, a U g {\displaystyle U{\mathfrak {g}}} -module). Considering R as a trivial representation of g {\displaystyle {\mathfrak {g}}} , one defines the cohomology groups
H n ( g ; M ) := E x t U g n ( R , M ) {\displaystyle \mathrm {H} ^{n}({\mathfrak {g}};M):=\mathrm {Ext} _{U{\mathfrak {g}}}^{n}(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 ↦ M g := { m ∈ M ∣ x m = 0 for all x ∈ g } . {\displaystyle 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
H n ( g ; M ) := T o r n U g ( R , M ) {\displaystyle \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 ↦ M g := M / g M . {\displaystyle 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.
Chevalley–Eilenberg complex Let g {\displaystyle {\mathfrak {g}}} be a Lie algebra over a field k {\displaystyle k} , with a left action on the g {\displaystyle {\mathfrak {g}}} -module M {\displaystyle M} . The elements of the Chevalley–Eilenberg complex
H o m k ( Λ ∙ g , M ) {\displaystyle \mathrm {Hom} _{k}(\Lambda ^{\bullet }{\mathfrak {g}},M)}
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