nLab
Chevalley-Eilenberg algebra

Idea

The Chevalley-Eilenberg algebra $\mathrm{CE}(𝔤)$ of a Lie algebra is a differential graded algebra of elements dual to $𝔤$ whose differential encodes the Lie bracket on $𝔤$.

The cochain cohomology of the underlying cochain complex is the Lie algebra cohomology of $𝔤$.

This generalizes to a notion of Chevalley-Eilenberg algebra for $𝔤$ an L-∞-algebra, a Lie algebroid and generally an ∞-Lie algebroid.

Grading conventions

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

$$\begin{array}{rl}{\wedge }^{•}V& :=\mathrm{Sym}(V[1])\\ & =k\oplus (V_0)\oplus (V_1\oplus V_0\wedge V_0)\oplus (V_2\oplus V_1\otimes V_0\oplus V_0\wedge V_0\wedge V_0)\oplus \cdots \end{array}$$\begin{aligned} \wedge^\bullet V &:= Sym(V[1]) \\ & = k \oplus (V_0) \oplus (V_1 \oplus V_0 \wedge V_0) \oplus (V_2 \oplus V_1 \otimes V_0 \oplus V_0 \wedge V_0 \wedge V_0) \oplus \cdots \end{aligned}

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 ${𝔤}^{*}$ of the vector space underlying a Lie algebra, we regard here as a $\mathbb{N}$-graded vector space in degree 0. For such, ${\wedge }^{•}{𝔤}^{*}$ is the ordinary Grassmann algebra over ${𝔤}^{*}$, where elements of ${𝔤}^{*}$ are generators of degree 1.

Of Lie algebras

Definition

The Chevalley-Eilenberg algebra $\mathrm{CE}(𝔤)$ of a finite dimensional Lie algebra $𝔤$ is the semifree graded-commutative dg-algebra whose underlying graded algebra is the Grassmann algebra

$${\wedge }^{•}{𝔤}^{*}=k\oplus {𝔤}^{*}\oplus ({𝔤}^{*}\wedge {𝔤}^{*})\oplus \cdots$$\wedge^\bullet \mathfrak{g}^* = k \oplus \mathfrak{g}^* \oplus (\mathfrak{g}^* \wedge \mathfrak{g}^* ) \oplus \cdots

(with the $n$th skew-symmetrized power in degree $n$)

and whose differential $d$ (of degree +1) is on ${𝔤}^{*}$ the dual of the Lie bracket

$$d{\mid }_{{𝔤}^{*}}:=[-,-{]}^{*}:{𝔤}^{*}\to {𝔤}^{*}\wedge {𝔤}^{*}$$d|_{\mathfrak{g}^*} := [-,-]^* : \mathfrak{g}^* \to \mathfrak{g}^* \wedge \mathfrak{g}^*

extended uniquely as a graded derivation on ${\wedge }^{•}{𝔤}^{*}$.

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 ${𝔤}^{*}$ and let $\{C^a{}_{bc}\}$ be the structure constants of the Lie bracket in that basis, then the action of the differential on the basis generators is

$$d{t}^a=-\frac{1}{2}{C}^a{}_{bc}{t}^b\wedge t^c,$$d t^a = - \frac{1}{2} C^a{}_{b c} t^b \wedge t^c \,,

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.,

Properties

One observes that for $𝔤$ a vector space, the graded-commutative dg-algebra structures on ${\wedge }^{•}{𝔤}^{*}$ are precisely in bijection with Lie algebra structures on $𝔤$: the dual of the restriction of $d$ to ${𝔤}^{*}$ 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 $𝔤\to 𝔥$ are precisely in bijection with morphisms of dg-algebras $\mathrm{CE}(𝔤)\leftarrow \mathrm{CE}(𝔥)$. And the $\mathrm{CE}$-construction is functorial.

Therefore, if we write ${\mathrm{dgAlg}}_{\mathrm{sf},1}$ 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

$$\mathrm{LieAlg}\stackrel{\mathrm{CE}(-)}{\underset{\simeq }{\to }}({\mathrm{dgAlg}}_{\mathrm{sf},1}{)}^{\mathrm{op}},$$LieAlg \stackrel{CE(-)}{\underset{\simeq}{\to}} (dgAlg_{sf,1})^{op} \,,

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 $𝔤$ are also in bijection with the structure of a differential graded coalgebra $({\vee }^{•}𝔤,D)$ on the free graded-co-commutative coalgebra ${\vee }^{•}𝔤$ on $𝔤$ with $D$ a derivation of degree -1 squaring to 0.

The relation between the differentials is simply dualization

$$({\vee }^{•}𝔤,D)\leftrightarrow ({\wedge }^{•}{𝔤}^{*},d)$$(\vee^\bullet \mathfrak{g}, D) \leftrightarrow (\wedge^\bullet \mathfrak{g}^* , d )

where for each $\omega \in {\wedge }^{•}{𝔤}^{*}$ we have

$$d\omega =\omega (D(-)).$$d \omega = \omega(D(-)) \,.

Of $L_{\mathrm{\infty }}$-algebras

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 ${𝔤}^{*}$ to the Grassmann algebra over a graded vector space.

If $𝔤$ is a graded vector space, then a differential $D$ of degree -1 squaring to 0 on ${\vee }^{•}𝔤$ is precisely the same as equipping $𝔤$ with the structure of an L-∞ algebra.

Dually, this corresponds to a general semifree dga

$$\mathrm{CE}(𝔤):=({\wedge }^{•}{𝔤}^{*},d=D^{*}).$$CE(\mathfrak{g}) := (\wedge^\bullet \mathfrak{g}^*, d = D^*) \,.

This we may usefully think of as the Chevalley-Eilenberg algebra of the $L_{\mathrm{\infty }}$-algebra $𝔤$.

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 $L_{\mathrm{\infty }}$-algebras.

Of Lie algebroids

For $𝔞$ a Lie algebroid given as

• a vector bundle $E\to X$

• with anchor map $\rho :E\to TX$

• and bracket $[-,-]:\Gamma (E){\wedge }_{C^{\mathrm{\infty }}(X)}\Gamma (E)\to \Gamma (E)$

the corresponding Chevalley-Eilenberg algebra is

$$\mathrm{CE}(𝔞):=({\wedge }_{C^{\mathrm{\infty }}(X)}^{•}\Gamma (E{)}^{*},d),$$CE(\mathfrak{a}) := (\wedge^\bullet_{C^\infty(X)} \Gamma(E)^*, d) \,,

where now the tensor products and dualization is over the ring $C^{\mathrm{\infty }}(X)$ of smooth functions on the base space $X$ (with values in the real numbers). The differential $d$ is given by the formula

$$(d\omega )(e_0,\cdots ,e_n)=\sum _{\sigma \in \mathrm{Shuff}(1,n)}\mathrm{sgn}(\sigma )\rho (e_{\sigma (0)})(\omega (e_{\sigma (1)},\cdots ,e_{\sigma (n)}))+\sum _{\sigma \in \mathrm{Shuff}(2,n-1)}\mathrm{sign}(\sigma )\omega ([e_{\sigma (0)},e_{\sigma (1)}],e_{\sigma (2)},\cdots ,e_{\sigma (n)}),$$(d\omega)(e_0, \cdots, e_n) = \sum_{\sigma \in Shuff(1,n)} sgn(\sigma) \rho(e_{\sigma(0)})(\omega(e_{\sigma(1)}, \cdots, e_{\sigma(n)})) + \sum_{\sigma \in Shuff(2,n-1)} sign(\sigma) \omega([e_{\sigma(0)},e_{\sigma(1)}],e_{\sigma(2)}, \cdots, e_{\sigma(n)} ) \,,

for all $\omega \in {\wedge }_{C^{\mathrm{\infty }}(X)}^n\Gamma (E{)}^{*}$ and $(e_i\in \Gamma (E))$, where $\mathrm{Shuff}(p,q)$ denotes the set of $(p,q)$-shuffles $\sigma$ and $\mathrm{sgn}(\sigma )$ the signature $\in \{\pm 1\}$ of the corresponding permutation.

For $X=*$ the point we have that $𝔞$ 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).

Of $\mathrm{\infty }$-Lie algebroids

See ∞-Lie algebroid.

Examples

Of abelian Lie $n$-algebras

The CE-algebra of the Lie algebra of the circle group $𝔲(1)$ is the graded-commutative dg-algebra on a single generator in degree 1 with vanishing differential.

More generally, the $L_{\mathrm{\infty }}$-algebra $b^n𝔲(1)$ is the one whose CE algebra is the commutative dg-algebra with a single generator in degree $n+1$ and vanishing differential.

Of $\mathrm{𝔰𝔲}(2)$

The CE-algebra of $\mathrm{𝔰𝔲}(2)$ has two generators $x,y,z$ in degree one and differential

$$d{x}_1=x_2\wedge x_3$$d x_1 = x_2 \wedge x_3

and cyclically.

Of the tangent Lie algebroid $TX$

For $X$ a smooth manifold and $TX$ its tangent Lie algebroid, the corresponding CE-algebra is the de Rham algebra of $X$.

$$\mathrm{CE}(TX)=({\wedge }_{C^{\mathrm{\infty }}(X)}^{•}\Gamma (T^{*}X),d_{\mathrm{dR}}).$$CE(T X) = (\wedge^\bullet_{C^\infty(X)} \Gamma(T^* X), d_{dR}) \,.

For $(v_i\in \Gamma (TX))$ vector fields and $\omega \in {\Omega }^n={\wedge }_{C^{\mathrm{\infty }}(X)}^n\Gamma (TX{)}^{*}$ a differential form of degree $n$, the formula for the CE-differential

$$(d\omega )(v_0,\cdots ,v_n)=\sum _{\sigma \in \mathrm{Sh}(1,n)}\mathrm{sgn}(\sigma )v_{\sigma (0)}(\omega (v_{\sigma (1)},\cdots ,v_{\sigma (n)}))+\sum _{\sigma \in \mathrm{Shuff}(2,n-1)}\mathrm{sgn}(\sigma )\omega ([v_{\sigma (0)},v_{\sigma (1)}],v_{\sigma (2)},\cdots ,v_{\sigma (n)}),$$(d\omega)(v_0, \cdots, v_n) = \sum_{\sigma \in Sh(1,n)} sgn(\sigma) v_{\sigma(0)}(\omega(v_{\sigma(1)}, \cdots, v_{\sigma(n)})) + \sum_{\sigma \in Shuff(2,n-1)} sgn(\sigma) \omega([v_{\sigma(0)},v_{\sigma(1)}],v_{\sigma(2)}, \cdots, v_{\sigma(n)} ) \,,

is indeed that for the de Rham differential.

Of the string Lie 2-algebra

For $𝔤$ a semisimple Lie algebra with binary invariant polynomial $⟨-,-⟩$ – the Killing form – , the CE-algebra of the string Lie 2-algebra is

$$\mathrm{CE}(\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤})=({\wedge }^{•}({𝔤}^{+}\oplus {ℝ}^{*}[1]),d_{\mathrm{string}})$$CE(\mathfrak{string}) = (\wedge^\bullet( \mathfrak{g}^+ \oplus \mathbb{R}^*[1]), d_{string})

where the differential restricted to ${𝔤}^{*}$ is $[-,-{]}^{*}$ while on the new generator $b$ spanning ${ℝ}^{*}[1]$ is it

$$db=⟨-,[-,-]⟩\in {\wedge }^3{𝔤}^{*}.$$d b = \langle -, [-,-]\rangle \in \wedge^3 \mathfrak{g}^* \,.

Weil algebra

For $𝔤$ a Lie algebra, the CE-algebra of the Lie 2-algebra given by the differential crossed module $(𝔤\stackrel{\mathrm{Id}}{\to }𝔤)$ is the Weil algebra $W(𝔤)$ of $𝔤$

$$\mathrm{CE}(𝔤\stackrel{\mathrm{Id}}{\to }𝔤)=W(𝔤).$$CE(\mathfrak{g} \stackrel{Id}{\to} \mathfrak{g}) = W(\mathfrak{g}) \,.

Lie algebra cohomology

Lie algebra cohomology of a $k$-Lie algebra $𝔤$ with coefficients in the left $𝔤$-module $M$ is defined as $H_{\mathrm{Lie}}^{*}(𝔤,M)={\mathrm{Ext}}_{U𝔤}^{*}(k,M)$. It can be computed as ${\mathrm{Hom}}_{𝔤}(V(𝔤),M)$ (a similar story is for Lie algebra homology) where $V(𝔤)=U(𝔤)\otimes {\Lambda }^{*}(𝔤)$ is the Chevalley-Eilenberg chain complex. If $𝔤$ is finite-dimensional over a field then ${\mathrm{Hom}}_{𝔤}(V(𝔤),k)=\mathrm{CE}(𝔤)={\Lambda }^{*}{𝔤}^{*}$ 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 $𝔤$ with values in the trivial module $ℝ$ is a morphism of L-∞ algebras

$$𝔤\to b^{n-1}𝔲(1).$$\mathfrak{g} \to b^{n-1} \mathfrak{u}(1) \,.

In terms of CE-algebras this is a dg-algebra morphism

$$\mathrm{CE}(𝔤)\leftarrow \mathrm{CE}(b^{n-1}𝔲(1)).$$CE(\mathfrak{g}) \leftarrow CE(b^{n-1}\mathfrak{u}(1)) \,.

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 $\mathrm{CE}(𝔤)$, i.e. an element $\omega \in {\wedge }^n{𝔤}^{*}$ which is closed under the CE-differential

$$d_{\mathrm{CE}}\omega =0.$$d_{CE} \omega = 0 \,.

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 $𝔤\to b^{n-1}𝔲(1)$.

BRST complex

In physics, the Chevalley-Eilenberg algebra $\mathrm{CE}(𝔤,N)$ 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 $\in {𝔤}^{*}$ 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

Related concepts

• ∞-Lie algebra cohomology

• Chevalley-Eilenberg algebra

• Weil algebra

• invariant polynomial

References

An elementary introduction for CE-algebras of Lie algebras is at the beginning of

• J. A. de Azcarraga, J. M. Izquierdo, J. C. Perez Bueno, An introduction to some novel applications of Lie algebra cohomology and physics (arXiv)

More details are in section 6.7 of

• J. A. de Azcárraga, José M. Izquierdo, Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics , Cambridge monographs of mathematical physics, (1995)

See also almost any text on Lie algebra cohomology (see the list of references there).