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Chevalley-Eilenberg algebra

Context

\infty-Lie theory

∞-Lie theory

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 \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 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 two 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

  • 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).

Revised on September 2, 2013 19:37:04 by Urs Schreiber (89.204.139.41)