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Chevalley-Eilenberg algebra

Context

\infty-Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

Idea

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

The cochain cohomology of the underlying cochain complex is the Lie algebra cohomology of 𝔤\mathfrak{g}.

This generalizes to a notion of Chevalley-Eilenberg algebra for 𝔤\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 VV an \mathbb{N}-graded vector space we write

V :=Sym(V[1]) =k(V 0)(V 1V 0V 0)(V 2V 1V 0V 0V 0V 0) \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 VV up in degree by one.

Here the elements in the nnth term in parenthesis are in degree nn.

A plain vector space, such as the dual 𝔤 *\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, 𝔤 *\wedge^\bullet \mathfrak{g}^* is the ordinary Grassmann algebra over 𝔤 *\mathfrak{g}^*, where elements of 𝔤 *\mathfrak{g}^* are generators of degree 1.

Of Lie algebras

Definition

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

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

(with the nnth skew-symmetrized power in degree nn)

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

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

extended uniquely as a graded derivation on 𝔤 *\wedge^\bullet \mathfrak{g}^*.

That this differential indeed squares to 0, dd=0d \circ d = 0, is precisely the fact that the Lie bracket satisfies the Jacobi identity.

If we choose a dual basis {t a}\{t^a\} of 𝔤 *\mathfrak{g}^* and let {C a bc}\{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

dt a=12C a bct bt 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\mathfrak{g} a vector space, the graded-commutative dg-algebra structures on 𝔤 *\wedge^\bullet \mathfrak{g}^* are precisely in bijection with Lie algebra structures on 𝔤\mathfrak{g}: the dual of the restriction of dd to 𝔤 *\mathfrak{g}^* defines a skew-linear bracket and the condition d 2=0d^2 = 0 holds if and only if that bracket satisfies the Jacobi identity.

Moreover, morphisms if Lie algebras 𝔤𝔥\mathfrak{g} \to \mathfrak{h} are precisely in bijection with morphisms of dg-algebras CE(𝔤)CE(𝔥)CE(\mathfrak{g}) \leftarrow CE(\mathfrak{h}). And the CECE-construction is functorial.

Therefore, if we write dgAlg sf,1dgAlg_{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

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

The relation between the differentials is simply dualization

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

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

dω=ω(D()). d \omega = \omega(D(-)) \,.

Of L 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 𝔤 *\mathfrak{g}^* to the Grassmann algebra over a graded vector space.

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

Dually, this corresponds to a general semifree dga

CE(𝔤):=( 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 L_\infty-algebra 𝔤\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 L_\infty-algebras.

Of Lie algebroids

For 𝔞\mathfrak{a} a Lie algebroid given as

  • a vector bundle EXE\to X

  • with anchor map ρ:ETX\rho : E \to T X

  • and bracket [,]:Γ(E) Γ(E)Γ(E)[-,-] \;\colon\; \Gamma(E)\wedge_{\mathbb{R}} \Gamma(E) \to \Gamma(E)

the corresponding Chevalley-Eilenberg algebra is

CE(𝔞):=( C (X) Γ(E) *,d), 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 (X)C^\infty(X) of smooth functions on the base space XX (with values in the real numbers). The differential dd is given by the formula

(dω)(e 0,,e n)= σShuff(1,n)sgn(σ)ρ(e σ(0))(ω(e σ(1),,e σ(n)))+ σShuff(2,n1)sign(σ)ω([e σ(0),e σ(1)],e σ(2),,e σ(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 ω C (X) nΓ(E) *\omega \in \wedge^n_{C^\infty(X)} \Gamma(E)^* and (e iΓ(E))(e_i \in \Gamma(E)), where Shuff(p,q)Shuff(p,q) denotes the set of (p,q)(p,q)-shuffles σ\sigma and sgn(σ)sgn(\sigma) the signature {±1}\in \{\pm 1\} of the corresponding permutation.

For X=*X = * the point we have that 𝔞\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 nn-algebras

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

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

Of 𝔰𝔲(2)\mathfrak{su}(2)

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

dx 1=x 2x 3 d x_1 = x_2 \wedge x_3

and cyclically.

Of the tangent Lie algebroid TXT X

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

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

For (v iΓ(TX))(v_i \in \Gamma(T X)) vector fields and ωΩ n= C (X) nΓ(TX) *\omega \in \Omega^n = \wedge^n_{C^\infty(X)} \Gamma(T X)^* a differential form of degree nn, the formula for the CE-differential

(dω)(v 0,,v n)= σSh(1,n)sgn(σ)v σ(0)(ω(v σ(1),,v σ(n)))+ σShuff(2,n1)sgn(σ)ω([v σ(0),v σ(1)],v σ(2),,v σ(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\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(𝔰𝔱𝔯𝔦𝔫𝔤)=( (𝔤 + *[1]),d string) CE(\mathfrak{string}) = (\wedge^\bullet( \mathfrak{g}^+ \oplus \mathbb{R}^*[1]), d_{string})

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

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

Weil algebra

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

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

Lie algebra cohomology

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

𝔤b n1𝔲(1). \mathfrak{g} \to b^{n-1} \mathfrak{u}(1) \,.

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

CE(𝔤)CE(b n1𝔲(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 nn and vanishing differential, such a morphism is precisely the same thing as a degree nn-element in CE(𝔤)CE(\mathfrak{g}), i.e. an element ω n𝔤 *\omega \in \wedge^n \mathfrak{g}^* which is closed under the CE-differential

d CEω=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 𝔤b n1𝔲(1)\mathfrak{g} \to b^{n-1}\mathfrak{u}(1).

BRST complex

In physics, the Chevalley-Eilenberg algebra CE(𝔤,N)CE(\mathfrak{g}, N) of the action of a Lie algebra or L-∞ algebra of a gauge group GG on space NN of fields is called the BRST complex.

In this context

  • the generators in NN in degree 0 are called fields;

  • the generators 𝔤 *\in \mathfrak{g}^* in degree 11 are called ghosts;

  • the generators in degree 22 are called ghosts of ghosts;

  • etc.

If NN is itself a chain complex, then this is called a BV-BRST complex

References

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

More details are in of

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

Revised on October 5, 2017 17:39:06 by Urs Schreiber (77.56.177.247)