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[1] 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.
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.
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.
More generally one uses a similar construction to define Lie algebra cohomology with coefficients in a module.
does not necessarily reproduce the de Rham cohomology of
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.
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.
be a Lie algebra over a field
The elements of the Chevalley–Eilenberg complex are called cochains from
is finitely generated as vector space, the Chevalley–Eilenberg complex is canonically isomorphic to the tensor product
denotes the dual vector space of
denote the left action of
according to the graded Leibniz rule, the nilpotency condition
following from the Lie algebra homomorphism from
where the caret signifies omitting that argument.
, the Chevalley–Eilenberg complex may also be canonically identified with the space of left-invariant forms with values in
The Chevalley–Eilenberg differential may then be thought of as a restriction of the covariant derivative on the trivial fiber bundle
is equipped with the trivial action of
to the subspace of left-invariant differential forms.
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
The second cohomology group is the space of equivalence classes of Lie algebra extensions of the Lie algebra by the module
Similarly, any element of the cohomology group
gives an equivalence class of ways to extend the Lie algebra
-algebra is a homotopy Lie algebra with nonzero terms only in degrees 0 through
, as mentioned earlier the Chevalley–Eilenberg complex coincides with the de-Rham complex for a corresponding compact Lie group.
carries the trivial action of
Finite dimensional, simple Lie algebras only have trivial central extensions: a proof is provided here.