nLab Chevalley-Eilenberg algebra

Idea

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.

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

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

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 $\mathfrak{g}^*$ of the vector space underlying a Lie algebra, is regarded as a $\mathbb{N}$-graded vector space in degree 0. The quantity $\wedge^\bullet \mathfrak{g}^*$ is the ordinary Grassmann algebra over $\mathfrak{g}^*$, where elements of $\mathfrak{g}^*$ are generators of degree 1.

Of Lie algebras

Definition

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

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

(with the $n$th skew-symmetrized power in degree $n$), and whose differential $d$ (of degree +1) is defined on $\mathfrak{g}^*$ as the dual of the Lie bracket

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

and 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

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

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 $\mathfrak{g}$ 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

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

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 $\mathfrak{g}$ 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

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

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

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

Of $L_\infty$-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 $\mathfrak{g}^*$ 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

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

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.

Of Lie algebroids

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

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

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

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

for all $\omega \in \wedge^n_{C^\infty(X)} \Gamma(E)^*$ 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).

Of $\infty$-Lie algebroids

See ∞-Lie algebroid.

Examples

Of abelian Lie $n$-algebras

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.

Of $\mathfrak{su}(2)$

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

$$d x_1 = x_2 \wedge x_3$$

and cyclically.

Of the tangent Lie algebroid $T X$

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

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

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

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

is indeed that for the de Rham differential.

Of the string Lie 2-algebra

For $\mathfrak{g}$ a semisimple Lie algebra with binary invariant polynomial $\langle -,-\rangle$ – the Killing form – , the CE-algebra of the string Lie 2-algebra is

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

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

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

Weil algebra

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 $\mathfrak{g}$

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

Lie algebra cohomology

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

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

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

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

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

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

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

BRST complex

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

Related concepts

• ∞-Lie algebra cohomology

• super L-∞ algebra

• Chevalley-Eilenberg algebra

  • differential graded-commutative algebra

  • differential graded-commutative superalgebra

• Weil algebra

• invariant polynomial

References

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

• José de Azcárraga, J. M. Izquierdo, J. C. Perez Bueno, An introduction to some novel applications of Lie algebra cohomology and physics (arXiv:physics/9803046)

More details are in of

• José de Azcárraga, José M. Izquierdo, section 6.7 of 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).