Lie algebra cohomology

Motivation

If 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. This can be replaced by the complex of equivariant differential forms, which can in turn 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 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.

Definition

Let g be a Lie algebra over a commutative ring R with universal enveloping algebra U g , and let M be a representation of g (equivalently, a U g -module). Considering R as a trivial representation of g , 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

The Lie algebra cohomology of the Lie algebra g over the field k , with values in the left g -module M can be computed using the Chevalley-Eilenberg complex H o m k ( Λ ∗ g , M ) . The n -cochains in this complex are the alternating k -multilinear functions f : Λ n g → M of n variables with values in M . The coboundary of an n -cochain is the ( n + 1 ) -cochain δ f given by

where the caret signifies omitting that argument.

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 Der of derivations modulo the space Ider of inner derivations

where a derivation is a map d from the Lie algebra to M such that

and is called inner if it is given by

for some a in M.

The second cohomology group

is the space of equivalence classes of Lie algebra extensions

of the Lie algebra by the module M.

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