group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
Lie algebra cohomology is the intrinsic notion of cohomology of Lie algebras.
There is a precise sense in which Lie algebras $\mathfrak{g}$ are infinitesimal Lie groups. Lie algebra cohomology is the restriction of the definition of Lie group cohomology to Lie algebras.
In ∞-Lie theory one studies the relation between the two via Lie integration.
Lie algebra cohomology generalizes to nonabelian Lie algebra cohomology and to ∞-Lie algebra cohomology.
There are several different but equivalent definitions of the cohomology of a Lie algebra.
The abelian cohomology of a $k$-Lie algebra $\mathfrak{g}$ with coefficients in the left $\mathfrak{g}$-module $M$ is defined as $H^*_{Lie}(\mathfrak{g},M) = Ext_{U\mathfrak{g}}^*(k,M)$ where $k$ is the ground field understood as a trivial module over the universal enveloping algebra $U\mathfrak{g}$. In particular it is a derived functor.
Before this approach was advanced in Cartan-Eilenberg’s Homological algebra, Lie algebra cohomology and homology were defined by Chevalley-Eilenberg with a help of concrete Koszul-type resolution which is in this case a cochain complex
where the first argument $U\mathfrak{g}\otimes_k \Lambda^* \mathfrak{g}$ is naturally equipped with a differential to start with (see below).
WHERE BELOW?
The first argument in the Hom, i.e. $U\mathfrak{g}\otimes_k \Lambda^* \mathfrak{g}$ is sometimes called the Chevalley-Eilenberg chain complex (cf. Weibel); the Chevalley-Eilenberg cochain complex is the whole thing, i.e.
If $M$ is a trivial module $k$ then $CE(\mathfrak{g}) := Hom_k(\Lambda^* \mathfrak{g},k)$ and if $\mathfrak{g}$ is finite-dimensional this equals $\Lambda^* \mathfrak{g}^*$ with an appropriate differential and the exterior multiplication gives it a dg-algebra structure.
As discussed at Chevalley-Eilenberg algebra, we may identify Lie algebras $\mathfrak{g}$ as the duals $CE(\mathfrak{g})$ of dg-algebras whose underlying graded algebra is the Grassmann algebra on the vector space $\mathfrak{g}^*$.
Similarly, a dg-algebra $CE(\mathfrak{h})$ whose underlying algebra is free on a graded vector space $\mathfrak{h}$ we may understand as exibiting an ∞-Lie algebra-structure on $\mathfrak{h}$.
Then a morphism $\mathfrak{g} \to \mathfrak{h}$ of these $\infty$-Lie algebras is by definition just a morphism $CE(\mathfrak{g}) \leftarrow CE(\mathfrak{h})$ of dg-algebras. Such a morphis may be thought of as a cocycle in nonabelian Lie algebra cohomology $H(\mathfrak{g}, \mathfrak{h})$.
Specifically, write $b^{n-1} \mathbb{R}$ for the line Lie n-algebra, the $\infty$-Lie algebra given by the fact that $CE(b^{n-1}\mathbb{R})$ has a single generator in degree $n$ and vanishing differential. Then a morphism
is a cocycle in the abelian Lie algebra cohomology $H^n(\mathfrak{g}, \mathbb{R})$. Notice that dually, by definition, this is a morphism of dg-algebras
Since on the right we only have a single closed degree-$n$ generator, such a morphism is precily a closed degree $n$-element
This way we recover the above definition of Lie algebra cohomology (with coefficient in the trivial module) in terms of the cochain complex cohomology of the CE-algebra.
The following lemma asserts that for semisimple Lie algebras $\mathfrak{g}$ only the cohomology $\mathfrak{g} \to b^{n-1} \mathbb{R}$ with coefficients in the trivial module is nontrivial.
Whiteheads lemma
For $\mathfrak{g}$ a finite dimensional semisimple Lie algebra over a field of characteristic 0, and for $V$ a non-trivial finite-dimensional irreducible representation, we have
Every invariant polynomial $\langle - \rangle \in W(\mathfrak{g})$ on a Lie algebra has a transgression to a cocycle on $\mathfrak{g}$. See ∞-Lie algebra cohomology for more.
For instance for $\mathfrak{g}$ a semisimple Lie algebra, there is the Killing form $\langle - ,- \rangle$. The corresponding 3-cocycle is
that is: the function that sends three Lie algebra elements $x, y, z$ to the number $\mu(x,y,z) = \langle x, [y,z]\rangle$.
On the super Poincare Lie algebra in dimension (10,1) there is a 4-cocycle
Every Lie algebra degree $n$ cocycle $\mu$ (with values in the trivial model) gives rise to an extension
In the language of ∞-Lie algebras this was observed in (BaezCrans Theorem 55).
In the dual dg-algebra language the extension is lust the relative Sullivan algebra
obtained by gluing on a rational $n$-sphere. By this kind of translation between familiar statements in rational homotopy theory dually into the language of ∞-Lie algebras many useful statements in ∞-Lie theory are obtained.
Examples
The string Lie 2-algebra is the extension of a semisimple Lie algebra induced by the canonical 3-cocycle coming from the Killing form.
The supergravity Lie 3-algebra is the extension of the super Poincare Lie algebra by a 4-cocycle.
An account of the standard theory of Lie algebra cohomology is for instance
in chapter V in vol III of
in section 6 of
with a brief summary in
chapter 7 of
See also
The cohomology of super Lie algebras is analyzed via normed division algebras in
John Baez, John Huerta, Division algebras and supersymmetry I (arXiv:0909.0551)
John Baez, John Huerta, Division algebras and supersymmetry II (arXiv:1003.34360)
See also division algebra and supersymmetry.
This subsumes some of the results in
The cohomology of the super Poincare Lie algebra in low dimensions $\leq 5$ is analyzed in
Supersymmetry algebra cohomology I: Definition and general structure J. Math. Phys.51:122302, 2010, arXiv
Supersymmetry algebra cohomology II: Primitive elements in 2 and 3 dimensions J. Math. Phys. 51 (2010) 112303 (arXiv)
Supersymmetry algebra cohomology III: Primitive elements in four and five dimensions (arXiv)
and in higher dimensions more generally in
The ∞-Lie algebra extensions $b^{n-2} \to \mathfrak{g}_\mu \to \mathfrak{g}$ induced by a degree $n$-cocycle are considered around theorem 55 in