∞-Lie theory (higher geometry)
The Chevalley-Eilenberg algebra $CE(\mathfrak{g})$ of a Lie algebra is a differential graded algebra of elements dual to $\mathfrak{g}$ whose differential encodes the Lie bracket on $\mathfrak{g}$.
The cochain cohomology of the underlying cochain complex is the Lie algebra cohomology of $\mathfrak{g}$.
This generalizes to a notion of Chevalley-Eilenberg algebra for $\mathfrak{g}$ an L-∞-algebra, a Lie algebroid and generally an ∞-Lie algebroid.
This differential-graded subject is somewhat notorious for a plethora of equivalent but different conventions on gradings and signs.
For the following we adopt the convention that for $V$ an $\mathbb{N}$-graded vector space we write
for the free graded-commutative algebra on the graded vector space obtained by shifting $V$ up in degree by one.
Here the elements in the $n$th term in parenthesis are in degree $n$.
A plain vector space, such as the dual $\mathfrak{g}^*$ of the vector space underlying a Lie algebra, we regard here as a $\mathbb{N}$-graded vector space in degree 0. For such, $\wedge^\bullet \mathfrak{g}^*$ is the ordinary Grassmann algebra over $\mathfrak{g}^*$, where elements of $\mathfrak{g}^*$ are generators of degree 1.
The Chevalley-Eilenberg algebra $CE(\mathfrak{g})$ of a finite dimensional Lie algebra $\mathfrak{g}$ is the semifree graded-commutative dg-algebra whose underlying graded algebra is the Grassmann algebra
(with the $n$th skew-symmetrized power in degree $n$)
and whose differential $d$ (of degree +1) is on $\mathfrak{g}^*$ the dual of the Lie bracket
extended uniquely as a graded derivation on $\wedge^\bullet \mathfrak{g}^*$.
That this differential indeed squares to 0, $d \circ d = 0$, is precisely the fact that the Lie bracket satisfies the Jacobi identity.
If we choose a dual basis $\{t^a\}$ of $\mathfrak{g}^*$ and let $\{C^a{}_{b c}\}$ be the structure constants of the Lie bracket in that basis, then the action of the differential on the basis generators is
where here and in the following a sum over repeated indices is implicit.
This has a more or less evident generalization to infinite-dimensional Lie algebras.,
One observes that for $\mathfrak{g}$ a vector space, the graded-commutative dg-algebra structures on $\wedge^\bullet \mathfrak{g}^*$ are precisely in bijection with Lie algebra structures on $\mathfrak{g}$: the dual of the restriction of $d$ to $\mathfrak{g}^*$ defines a skew-linear bracket and the condition $d^2 = 0$ holds if and only if that bracket satisfies the Jacobi identity.
Moreover, morphisms if Lie algebras $\mathfrak{g} \to \mathfrak{h}$ are precisely in bijection with morphisms of dg-algebras $CE(\mathfrak{g}) \leftarrow CE(\mathfrak{h})$. And the $CE$-construction is functorial.
Therefore, if we write $dgAlg_{sf,1}$ for the category whose objects are semifree dgas on generators in degree 1, we find that the construction of CE-algebras from Lie algebras constitutes a canonical equivalence of categories
where on the right we have the opposite category.
This says that in a sense the Chevalley-Eilenberg algebra is just another way of looking at (finite dimensional) Lie algebras.
There is an analogous statement not involving the dualization: Lie algebra structures on $\mathfrak{g}$ are also in bijection with the structure of a differential graded coalgebra $(\vee^\bullet \mathfrak{g}, D)$ on the free graded-co-commutative coalgebra $\vee^\bullet \mathfrak{g}$ on $\mathfrak{g}$ with $D$ a derivation of degree -1 squaring to 0.
The relation between the differentials is simply dualization
where for each $\omega \in \wedge^\bullet \mathfrak{g}^*$ we have
The equivalence between Lie algebras and differential graded algebras/coalgebras discussed above suggests a grand generalization by simply generalizing the Grassmann algebra over a vector space $\mathfrak{g}^*$ to the Grassmann algebra over a graded vector space.
If $\mathfrak{g}$ is a graded vector space, then a differential $D$ of degree -1 squaring to 0 on $\vee^\bullet \mathfrak{g}$ is precisely the same as equipping $\mathfrak{g}$ with the structure of an L-∞ algebra.
Dually, this corresponds to a general semifree dga
This we may usefully think of as the Chevalley-Eilenberg algebra of the $L_\infty$-algebra $\mathfrak{g}$.
So every commutative semifree dga (degreewise finite-dimensional) is the Chevaley-Eilenberg algebra of some L-∞ algebra of finite type.
This means that many constructions involving dg-algebras are secretly about ∞-Lie theory. For instance the Sullivan construction in rational homotopy theory may be interpreted in terms of Lie integration of $L_\infty$-algebras.
For $\mathfrak{a}$ a Lie algebroid given as
a vector bundle $E\to X$
with anchor map $\rho : E \to T X$
and bracket $[-,-] \;\colon\; \Gamma(E)\wedge_{\mathbb{R}} \Gamma(E) \to \Gamma(E)$
the corresponding Chevalley-Eilenberg algebra is
where now the tensor products and dualization is over the ring $C^\infty(X)$ of smooth functions on the base space $X$ (with values in the real numbers). The differential $d$ is given by the formula
for all $\omega \in \wedge^n_{C^\infty(X)} \Gamma(E)^*$ and $(e_i \in \Gamma(E))$, where $Shuff(p,q)$ denotes the set of $(p,q)$-shuffles $\sigma$ and $sgn(\sigma)$ the signature $\in \{\pm 1\}$ of the corresponding permutation.
For $X = *$ the point we have that $\mathfrak{a}$ is a Lie algebra and this definition reproduces the above definition of the CE-algebra of a Lie algebra (possibly up to an irrelevant global sign).
See ∞-Lie algebroid.
The CE-algebra of the Lie algebra of the circle group $\mathfrak{u}(1)$ is the graded-commutative dg-algebra on a single generator in degree 1 with vanishing differential.
More generally, the $L_\infty$-algebra $b^n \mathfrak{u}(1)$ is the one whose CE algebra is the commutative dg-algebra with a single generator in degree $n+1$ and vanishing differential.
The CE-algebra of $\mathfrak{su}(2)$ has three generators $x, y, z$ in degree one and differential
and cyclically.
For $X$ a smooth manifold and $T X$ its tangent Lie algebroid, the corresponding CE-algebra is the de Rham algebra of $X$.
For $(v_i \in \Gamma(T X))$ vector fields and $\omega \in \Omega^n = \wedge^n_{C^\infty(X)} \Gamma(T X)^*$ a differential form of degree $n$, the formula for the CE-differential
is indeed that for the de Rham differential.
For $\mathfrak{g}$ a semisimple Lie algebra with binary invariant polynomial $\langle -,-\rangle$ – the Killing form – , the CE-algebra of the string Lie 2-algebra is
where the differential restricted to $\mathfrak{g}^*$ is $[-,-]^*$ while on the new generator $b$ spanning $\mathbb{R}^*[1]$ is it
For $\mathfrak{g}$ a Lie algebra, the CE-algebra of the Lie 2-algebra given by the differential crossed module $(\mathfrak{g} \stackrel{Id}{\to} \mathfrak{g})$ is the Weil algebra $W(\mathfrak{g})$ of $\mathfrak{g}$
Lie algebra cohomology of a $k$-Lie algebra $\mathfrak{g}$ with coefficients in the left $\mathfrak{g}$-module $M$ is defined as $H^*_{Lie}(\mathfrak{g},M) = Ext_{U\mathfrak{g}}^*(k,M)$. It can be computed as $Hom_{\mathfrak{g}}(V(\mathfrak{g}),M)$ (a similar story is for Lie algebra homology) where $V(\mathfrak{g})=U(\mathfrak{g})\otimes\Lambda^*(\mathfrak{g})$ is the Chevalley-Eilenberg chain complex. If $\mathfrak{g}$ is finite-dimensional over a field then $Hom_{\mathfrak{g}}(V(\mathfrak{g}),k) = CE(\mathfrak{g}) = \Lambda^* \mathfrak{g}^*$ is the underlying complex of the Chevalley-Eilenberg algebra, i.e. the Chevalley-Eilenberg cochain complex with trivial coefficients.
A cocycle in degree n of the Lie algebra cohomology of a Lie algebra $\mathfrak{g}$ with values in the trivial module $\mathbb{R}$ is a morphism of L-∞ algebras
In terms of CE-algebras this is a dg-algebra morphism
Since by the above example the dg-algebra on he right has a single generator in degree $n$ and vanishing differential, such a morphism is precisely the same thing as a degree $n$-element in $CE(\mathfrak{g})$, i.e. an element $\omega \in \wedge^n \mathfrak{g}^*$ which is closed under the CE-differential
This is what one often sees as the definition of a cocycle in Lie algebra cohomology. However, from the general point of view of cohomology, it is better to think of the cocycle equivalently as the morphism $\mathfrak{g} \to b^{n-1}\mathfrak{u}(1)$.
In physics, the Chevalley-Eilenberg algebra $CE(\mathfrak{g}, N)$ of the action of a Lie algebra or L-∞ algebra of a gauge group $G$ on space $N$ of fields is called the BRST complex.
In this context
the generators in $N$ in degree 0 are called fields;
the generators $\in \mathfrak{g}^*$ in degree $1$ are called ghosts;
the generators in degree $2$ are called ghosts of ghosts;
etc.
If $N$ is itself a chain complex, then this is called a BV-BRST complex
Chevalley-Eilenberg algebra
An elementary introduction for CE-algebras of Lie algebras is at the beginning of
More details are in section 6.7 of
See also almost any text on Lie algebra cohomology (see the list of references there).