nLab
Weil algebra

There are two different concepts called Weil algebra. This entry is about the notion of Weil algebra in Lie theory. For the notion in infinitesimal geometry see infinitesimally thickened point/Artin 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 notion of Weil algebra is ordinarily defined for a Lie algebra 𝔤\mathfrak{g}. It may be understood as the Chevalley-Eilenberg algebra of the tangent Lie 2-algebra T𝔤T \mathfrak{g} or inn(𝔤)inn(\mathfrak{g}) of 𝔤\mathfrak{g}, generalizing the notion of tangent Lie algebroid TXT X from a 0-truncated Lie algebroid XX (a smooth manifold) to the one-obeject Lie algebroid 𝔤\mathfrak{g}.

Generally, for every Lie-∞-algebroid 𝔞\mathfrak{a} one may define the corresponding tangent Lie-\infty-algebroid T𝔞T \mathfrak{a}, whose Chevalley-Eilenberg algebra may be called the Weil algebra of 𝔞\mathfrak{a}:

W(𝔞)=CE(T𝔞). W(\mathfrak{a}) = CE(T \mathfrak{a}) \,.

Weil algebra of a Lie algebra

Let 𝔤\mathfrak{g} be a finite-dimensional Lie algebra. The Weil algebra W(𝔤)W(\mathfrak{g}) of 𝔤\mathfrak{g} is

  • the graded Grassmann algebra generated from the dual vector space 𝔤 *\mathfrak{g}^* together with another copy of 𝔤 *\mathfrak{g}^* shifted in degree

    W(𝔤):= (𝔤 *𝔤 *[1]) W(\mathfrak{g}) := \wedge^\bullet (\mathfrak{g}^* \oplus \mathfrak{g}^*[1])
  • equipped with a derivation d:W(𝔤)W(𝔤)d : W(\mathfrak{g}) \to W(\mathfrak{g}) that makes this a dg-algebra, defined by the fact that on 𝔤 *\mathfrak{g}^* it acts as the differential of the Chevalley-Eilenberg algebra of 𝔤\mathfrak{g} plus the degree shift morphism 𝔤 *𝔤 *\mathfrak{g}^* \to \mathfrak{g}^*.

This Weil algebra has trivial cohomology everywhere (except in degree 0 of course) and sits in a sequence

CE(𝔤)W(𝔤)inv(𝔤) CE(\mathfrak{g}) \leftarrow W(\mathfrak{g}) \leftarrow inv(\mathfrak{g})

with the Chevalley-Eilenberg algebra of 𝔤\mathfrak{g} and its algebra of invariant polynomials on 𝔤\mathfrak{g}. This may be understood as a model for the sequence of algebras of differential forms on the universal G-bundle

GGG. G \to \mathcal{E}G \to \mathcal{B}G \,.

As such, the Weil algebra plays a crucial role in the study of the Lie algebra cohomology of 𝔤\mathfrak{g}.

Definition

We first consider Weil algebras of L-∞ algebras, then more generally of L-∞ algebroids.

We use the notation and grading conventions that are described in detail at Chevalley-Eilenberg algebra.

For L L_\infty-algebras

Let 𝔤\mathfrak{g} be an L-∞ algebra of finite type. By our grading conventions this means that the graded vector space 𝔤 *\mathfrak{g}^* obtained by degreewise dualization is in non-negative degree, and 1𝔤 *=𝔤 *[1]\wedge^1 \mathfrak{g}^* = \mathfrak{g}^*[1] is its shift up into positive degree.

A quick abstract way to characterize the Weil algebra of 𝔤\mathfrak{g} is as follows. Notice that there is a free functor/forgetful functor adjunction

(FU):dgAlgUFVect[] (F \dashv U) : dgAlg \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}} Vect[\mathbb{Z}]

between the category dgAlg of dg-algebras and the category of \mathbb{Z}-graded vector spaces (all over some fixed field). Notice that a free object is unique up to isomorphism .

Definition

The Weil algebra W(𝔤)W(\mathfrak{g}) is the unique representative of the free dg-algebra on 1𝔤 *\wedge^1 \mathfrak{g}^* for which the projection of graded vector spaces 1(𝔤 *𝔤 *[1]) 1𝔤 *\wedge^1(\mathfrak{g}^* \oplus \mathfrak{g}^*[1]) \to \wedge^1 \mathfrak{g}^* extended to a dg-algebra homomorphism W(𝔤)CE(𝔤)W(\mathfrak{g}) \to CE(\mathfrak{g})

We discuss below in the Properties section that this is equivalent to the following component-wise definition

Definition

The Weil algebra W(𝔤)W(\mathfrak{g}) is the semi-free dga whose underlying graded-commutative algebra is the exterior algebra

(𝔤 *𝔤 *[1]) \wedge^\bullet (\mathfrak{g}^* \oplus \mathfrak{g}^*[1])

on 𝔤 *\mathfrak{g}^* and a shifted copy of 𝔤 *\mathfrak{g}^*, and whose differential is the sum

d W(𝔤)=d CE(𝔤)+d d_{W(\mathfrak{g})} = d_{CE(\mathfrak{g})} + \mathbf{d}

of two graded derivations of degree +1 defined by

  • d\mathbf{d} acts by degree shift 𝔤 *𝔤 *[1]\mathfrak{g}^* \to \mathfrak{g}^*[1] on elements in 𝔤 *\mathfrak{g}^* and by 0 on elements of 𝔤 *[1]\mathfrak{g}^*[1];

  • d CE(𝔤)d_{CE(\mathfrak{g})} acts on unshifted elements in 𝔤 *\mathfrak{g}^* as the differential of the Chevalley-Eilenberg algebra of 𝔤\mathfrak{g} and is extended uniquely to shifted generators by graded-commutattivity

    [d CE(𝔤,d]=0 [d_{CE(\mathfrak{g}}, \mathbf{d}] = 0

    with d\mathbf{d}:

    d CE(𝔤)dω:=dd CE(𝔤)ω d_{CE(\mathfrak{g})} \mathbf{d} \omega := - \mathbf{d} d_{CE(\mathfrak{g})} \omega

    for all ω 1𝔤 *\omega \in \wedge^1 \mathfrak{g}^*.

For L L_\infty-algebroids

Where the Chevalley-Eilenberg algebra of an L-∞ algebra has in degree 0 the ground field, that of an L-∞ algebroid has more generally an algebra over a Lawvere theory. For L-∞ algebroids over smooth manifolds this is the algebra of smooth functions on a manifolds, regarded as a smooth algebra (C C^\infty-ring).

So let TT be a Fermat theory. Write TAlgT Alg for the corresponding category of algebra. There is a free functor/forgetful functor adjunction

(FU):TAlgUFCRing (F \dashv U) : T Alg \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}} CRing

to the category CRing of commutative Rings.

We need the facts that

  • a module over a TT-algebra AA is uniquely specified by its underlying module over U(A)U(A);

  • the universal derivation on a TT-algebra AA is the de Rham differential

    d dR:AΩ 1(A) d_{dR} : A \to \Omega^1(A)

with values in the AA-module of TT-Kähler differentials.

See the corresponding entries for more details. The second point means that for v:ANv : A \to N any TT-derivation on AA, there is a unique AA-module homomorphism

Ω (A)N \Omega^\bullet(A) \to N

such that the diagram

Ω (A) d dR v A v N \array{ && \Omega^\bullet(A) \\ & {}^{\mathllap{d_{dR}}}\nearrow & \downarrow^{\mathrlap{v}} \\ A &\stackrel{v}{\to}& N }

commutes.

Let now 𝔞\mathfrak{a} be an L-∞ algebroid with Chevalley-Eilenberg algebra considered as the following data;

  1. a graded commutative semifree dga CE(𝔞)CE(\mathfrak{a}) over the ground field;

  2. the structure of a TT-algebra on the associative algebra A:=CE(𝔞) 0A := CE(\mathfrak{a})_0 (over the ground field)

    such that d CE(𝔞):CE(𝔞) 0CE(𝔞) 1d_{CE(\mathfrak{a})} : CE(\mathfrak{a})_0 \to CE(\mathfrak{a})_1 is a derivation of TT-algebra modules.

By semi-freeness there exists a \mathbb{N}-graded vector space (𝔞 *) (\mathfrak{a}^*)^\bullet and an isomorphism

CE(𝔞)( A (𝔞 *),d CE(𝔞)). CE(\mathfrak{a}) \simeq (\wedge^\bullet_{A} (\mathfrak{a}^*), d_{CE(\mathfrak{a})}) \,.
Definition

The Weil algebra W(𝔞)W(\mathfrak{a}) of the L L_\infty-algebroid 𝔞\mathfrak{a} is the Chevalley-Eilenberg algebra of the L L_\infty-algebroid defined as follows

  • the TT-algebra AA in degree 0 is the same as that of 𝔄\mathfrak{A};

  • the underlying graded algebra is the exterior algebra on 𝔞 *\mathfrak{a}^* and a shifted copy 𝔞 *[1]\mathfrak{a}^*[1] as well as one copy of the Kähler differential module Ω 1\Omega^1 in lowest degree (though of as the shifted copy of AA itself)

    (Ω 1(A)(𝔞 *)𝔞 *[1]). \wedge^\bullet (\Omega^1(A) \oplus (\mathfrak{a}^*) \oplus \mathfrak{a}^*[1]) \,.
  • the differential is the sum

    d W(𝔞)=d CE(𝔞)+d d_{W(\mathfrak{a})} = d_{CE(\mathfrak{a})} + \mathbf{d}

    of two degree +1 graded derivations, where d CE(𝔞)d_{CE(\mathfrak{a})} and a\mathbf{a} are defined on 1𝔞 *𝔞 *[1]\wedge^1 \mathfrak{a}^* \oplus \mathfrak{a}^*[1] as above for L L_\infty-algebras and on AA itself d CE(𝔞)d_{CE(\mathfrak{a})} vanishes and d\mathbf{d} acts as the universal derivation

    d| A=d dR:AΩ 1(A). \mathbf{d}|_A = d_{\mathrm{dR}} : A \to \Omega^1(A) \,.

Properties

Free property

The main point of the definition is that the differential restricted to the original (unshifted) generators is the original differential plus the shift:

d W(𝔞)| 𝔞 *=d CE(𝔞)+d. d_{W(\mathfrak{a})} |_{\mathfrak{a}^*} = d_{CE(\mathfrak{a})} + \mathbf{d} \,.

By solving the condition d W(𝔞)d W(𝔞)=0d_{W(\mathfrak{a})} \circ d_{W(\mathfrak{a})} = 0 and using that d CE(𝔞)d CE(𝔞)=0d_{CE(\mathfrak{a})} d_{CE(\mathfrak{a})} = 0 this already fixes uniquely the differential d W(𝔞)d_{W(\mathfrak{a})}. To see this we only need to show that the value of d W(𝔞)(x)d_{W(\mathfrak{a})}(x) on a generator x=σ(t)𝔞 *[1]x=\sigma(t) \in \mathfrak{a}^*[1] is completely determined by d W(𝔞)| 𝔞 *d_{W(\mathfrak{a})}\vert_{\wedge^\bullet\mathfrak{a}^*}. One computes:

0 =d W(𝔞)(d W(𝔞)t) =d W(𝔞)(d CE(𝔞)t+σt) =σd CE(𝔞)t+d W(𝔞)x \begin{aligned} 0 & = d_{W(\mathfrak{a})}(d_{W(\mathfrak{a})} t) \\ & = d_{W(\mathfrak{a})}(d_{CE(\mathfrak{a})}t + \sigma t) \\ & = \sigma d_{CE(\mathfrak{a})} t + d_{W(\mathfrak{a}) } x \end{aligned}

and hence

d W(𝔞)x=σd CE(𝔞)σ 1(x). d_{W(\mathfrak{a})} x = - \sigma d_{CE(\mathfrak{a})} \sigma^{-1} (x) \,.

This implies the following universal freeness property:

Proposition

Let 𝔤\mathfrak{g} be an L L_\infty-algebra. Morphisms of dgdg-algebras W(𝔤)AW(\mathfrak{g}) \to A are in natural bijection to morphisms of graded vector spaces 𝔤 *A\mathfrak{g}^* \to A.

Proof

Forgetting the differential, W(𝔤)W(\mathfrak{g}) is the free graded-commutative algebra generated by (a shifted copy of) 𝔤 *\mathfrak{g}^* and 𝔤 *[1]\mathfrak{g}^*[1]. Therefore,

Hom dgca(W(𝔤),A)Hom gca(W(𝔤),A)=Hom grVect(𝔤 *,A)Hom grVect(𝔤 *[1],A). Hom_{dgca}(W(\mathfrak{g}),A)\subseteq Hom_{gca}(W(\mathfrak{g}),A)=Hom_{grVect}(\mathfrak{g}^*,A)\oplus Hom_{grVect}( \mathfrak{g}^*[1],A).

Projecting down to Hom grVect(𝔤 *,A)Hom_{grVect}(\mathfrak{g}^*,A), one obtains a natural map

Hom dgca(W(𝔤),A)Hom grVect(𝔤 *,A), Hom_{dgca}(W(\mathfrak{g}),A)\to Hom_{grVect}(\mathfrak{g}^*,A),

which is a bijection.

To prove injectivity, we just have to show that the restriction of a dgca morphism f:W(𝔤)Af:W(\mathfrak{g})\to A to 𝔤 *\mathfrak{g}^* determines the restriction of ff to 𝔤 *[1]\mathfrak{g}^*[1]. One has, for any x=σ(t)𝔤 *[1]x=\sigma(t)\in \mathfrak{g}^*[1],

f(x) =f(σ(t))=f(d W(𝔤)td CE(𝔤)t) =d Af(t)f(d CE(𝔤)t). \begin{aligned} f(x)&=f(\sigma(t))=f(d_{W(\mathfrak{g})}t-d_{CE(\mathfrak{g})}t)\\ &=d_A f(t)- f(d_{CE(\mathfrak{g})}t). \end{aligned}

Since d CE(𝔤)(t)d_{CE(\mathfrak{g})}(t) lies in the sub-gca of W(𝔤)W(\mathfrak{g}) generated by 𝔤 *\mathfrak{g}^*, the element f(d CE(𝔤)(t))f(d_{CE(\mathfrak{g})}(t)), and therefore f(x)f(x), is determined by f| 𝔤 *f\vert_{\mathfrak{g}^*}.

Next we show surjectivity, i.e. that every morphism of graded vector spaces ϕ:𝔤 *A\phi:\mathfrak{g}^*\to A can be extended to a dgca morphism f:W(𝔤)Af:W(\mathfrak{g})\to A. Denote by f 0: 𝔤 *Af_0: \wedge^\bullet \mathfrak{g}^*\to A the extension of ϕ\phi to a graded commutative algebra morphism, and let ψ:𝔤 *[1]A\psi:\mathfrak{g}^*[1]\to A be the graded vector space morphism defined by

ψ(x)=d Aϕ(t)f 0d CE(𝔤)(t), \psi(x)=d_A \phi(t)-f_0d_{CE(\mathfrak{g})}(t),

for any x=σ(t)𝔤 *[1]x=\sigma(t)\in \mathfrak{g}^*[1]. The graded vector space morphism ϕ+ψ:𝔤 *𝔤 *[1]A\phi+\psi:\mathfrak{g}^*\oplus\mathfrak{g}^*[1]\to A extends to a commutative graded algebra f:W(𝔤)Af:W(\mathfrak{g})\to A, whose restriction to 𝔤 *\mathfrak{g}^* is ϕ\phi. We want to show that ff is actually a dgca morphism. We only need to test commutativity with the differentials on generators t𝔤 *t\in \mathfrak{g}^* and x=σ(t)𝔤 *[1]x=\sigma(t)\in \mathfrak{g}^*[1]. We have

d Af(t)=d Aϕ(t)=ψ(σ(t))+f 0d CE(𝔤)(t)=f(σ(t))+fd CE(𝔤)(t)=fd W(𝔤)(t), d_A f(t)=d_A\phi(t)=\psi(\sigma(t))+f_0d_{CE(\mathfrak{g})}(t)=f(\sigma(t))+ f d_{CE(\mathfrak{g})}(t)=f d_{W(\mathfrak{g})}(t),

which in particular implies that d Af| 𝔤 *=fd W(𝔤)| 𝔤 *d_A f\vert_{\wedge^\bullet \mathfrak{g}^*}=f d_{W(\mathfrak{g})}\vert_{\wedge^\bullet \mathfrak{g}^*}, and

d Af(x)=d Aψ(x)=d Af 0d CE(𝔤)(t)=d Af(d CE(𝔤)(t)). d_A f(x)= d_A \psi(x) = -d_A f_0d_{CE(\mathfrak{g})}(t)=-d_A f (d_{CE(\mathfrak{g})}(t)).

Since d CE(𝔤)(t) 𝔤 *d_{CE(\mathfrak{g})}(t)\in \wedge^\bullet \mathfrak{g}^*, we obtain

d Af(x)=fd W(𝔤)(d CE(𝔤)(t))=fd W(𝔤)(d W(𝔤)(t)x)=fd W(𝔤)(x). d_A f(x)= -f d_{W(\mathfrak{g})} (d_{CE(\mathfrak{g})}(t))= -f d_{W(\mathfrak{g})}(d_{W(\mathfrak{g})}(t)-x)=f d_{W(\mathfrak{g})}(x).
Example

For A=CE(𝔤)A=CE(\mathfrak{g}) the Chevalley-Eilenberg algebra of 𝔤\mathfrak{g}, the inclusion 𝔤 *CE(𝔤)\mathfrak{g}^*\hookrightarrow CE(\mathfrak{g}) induces a canonical surjective dgca morphism W(𝔤)CE(𝔤)W(\mathfrak{g})\to CE(\mathfrak{g}). This is the identity on the unshifted generators, and 0 on the shifted generators.

Example

For A=Ω (X)A = \Omega^\bullet(X) the de Rham complex of a smooth manifold XX, we have that

Hom dgAlg(W(𝔤),Ω (X))=(Ω (X)𝔤) 1 Hom_{dgAlg}(W(\mathfrak{g}), \Omega^\bullet(X)) = (\Omega^\bullet(X) \otimes \mathfrak{g})^1

is the collection of total degree 1 differential forms with values in the \infty-Lie algebra 𝔤\mathfrak{g}.

A morphism of

(A,F A):W(𝔤)Ω (X) (A, F_A) : W(\mathfrak{g}) \to \Omega^\bullet(X)

sends the unshifted generators t at^a to differential forms A aA^a, which one thinks of as local connection forms, and sends the shifted generators σt a\sigma t^a to their curvature. The respect for the differential on the shifted generators is the Bianchi identity on these curvatures.

A morphism W(𝔤)Ω (X)W(\mathfrak{g}) \to \Omega^\bullet(X) encodes a collection of flat L L_\infty-algebra valued forms precisely if it factors by the canonical morphism W(𝔤)CE(𝔤)W(\mathfrak{g}) \to CE(\mathfrak{g}) from above through the Chevalley-Eilenberg algebra of 𝔤\mathfrak{g}.

The freeness property of the Weil algebra can be made more explicit by exhibiting a concrete isomorphism to the free dg-algebra on 𝔤 *\mathfrak{g}^*.

Definition

The canonical free dg-algebra on 𝔤 *\mathfrak{g}^* is

F(𝔤):= (𝔤 *𝔤 *[1],d F) F(\mathfrak{g}) := \wedge^\bullet( \mathfrak{g}^* \oplus \mathfrak{g}^*[1], d_F )

where the differential d fd_f is on the unshifted generators t𝔤 *t \in \mathfrak{g}^* the shift isomorphism σ:𝔤 *𝔤 *[1]\sigma : \mathfrak{g}^* \to \mathfrak{g}^*[1] extended as a derivation and vanishes on the shifted generators

d F:tσ(t), d_F : t \mapsto \sigma(t) \,,
d F:σ(t)0. d_F : \sigma(t) \mapsto 0 \,.

Or in other words, if 𝔤¯\bar \mathfrak{g} is the \infty-Lie algebra whose underlying graded vector space is that of 𝔤\mathfrak{g}, but all whose brackets vanish, then

F(𝔤)=W(𝔤¯). F(\mathfrak{g}) = W(\bar \mathfrak{g}) \,.

Notice the evident

Observation

The cochain cohomology of F(𝔤)F(\mathfrak{g}) vanishes in positive degree.

To see this, let K:=σ 1:F(𝔤)F(𝔤)K := \sigma^{-1} : F(\mathfrak{g}) \to F(\mathfrak{g}) be the degree down-shift isomorphism 𝔤 *[1]𝔤 *\mathfrak{g}^*[1] \to \mathfrak{g}^* extended as a graded derivation of degree -1, then

[d F(𝔤),K]=Id:F(𝔤)F(𝔤) [d_{F(\mathfrak{g})}, K] = Id : F(\mathfrak{g}) \to F(\mathfrak{g})

and hence for any ωF(𝔤)\omega \in F(\mathfrak{g}) such that d F(𝔤)ω=0d_{F(\mathfrak{g})} \omega = 0 we have ω=d F(𝔤)Kω\omega = d_{F(\mathfrak{g})} K \omega.

Lemma

Given 𝔤\mathfrak{g}, there is an isomorphism of dg-algebras

f:F(𝔤)W(𝔤) f : F(\mathfrak{g}) \to W(\mathfrak{g})

given by

f:tt f : t \mapsto t
f:σ(t)d W(𝔤)t=d CE(𝔤)t+σ(t). f : \sigma(t) \mapsto d_{W(\mathfrak{g})} t = d_{CE(\mathfrak{g})} t + \sigma(t) \,.
Proof

It is clear that ff is a dg-algebra homomorphism. The inverse dg-algebra morphism is given on generators by

f 1:tt f^{-1} : t \mapsto t
f 1:σ(t)σ(t)d CE(𝔤)(t). f^{-1} : \sigma(t) \mapsto \sigma(t) - d_{CE(\mathfrak{g})}(t) \,.

Note that the isomorphism ff is precisely the dgca isomorphism induced between W(𝔤¯)W(\overline\mathfrak{g}) and W(𝔤)W(\mathfrak{g}) by the identity of 𝔤 *\mathfrak{g}^* as a graded vector spaces morphism 𝔤¯ *𝔤 *\overline{\mathfrak{g}}^*\to\mathfrak{g}^*.

Corollary

The cochain cohomology of the Weil algebra of an L L_\infty-algebra is trivial.

Remark

This means that homotopy-theoretically the Weil algebra is the point. Dually, the \infty-Lie algebra inn(𝔤)inn(\mathfrak{g}) is a model for the point. In fact, one can see that inn(𝔤)inn(\mathfrak{g}) is the universal principal ∞-bundle over 𝔤\mathfrak{g} in the canonical model for the (∞,1)-topos SynthDiff∞Grpd. In fact, it is a groupal model for universal principal ∞-bundles. This is discussed at ∞-Lie algebra cohomology.

As the CE-algebra of the L L_\infty-algebra of inner derivations

By the discussion at ∞-Lie algebra and Chevalley-Eilenberg algebra, we may identify the full subcategory of the opposite category dgAlg on commutative semi-free dgas in non-negative degree with that of ∞-Lie algebras/∞-Lie algebroids.

That means that the Weil algebra W(𝔤)W(\mathfrak{g}) of some L-∞ algebra 𝔤\mathfrak{g} is the Chevalley-Eilenberg algebra of another \infty-Lie algebra.

Definition

For any \infty-Lie algebra 𝔤\mathfrak{g} write inn(𝔤)inn(\mathfrak{g}) for the \infty-Lie algebra whose CE-algebra is W(𝔤)W(\mathfrak{g}):

CE(inn(𝔤)):=W(𝔤). CE(inn(\mathfrak{g})) := W(\mathfrak{g}) \,.

In the following we discuss these inner automorphism \infty-Lie algebras in more detail. (See section 6 of (SSSI)).

For an ordinary Lie algebra

Observation

For 𝔤\mathfrak{g} an ordinary Lie algebra the inner derivation Lie 2-algebra is the strict Lie 2-algebra given by the dg-Lie algebra

inn(𝔤)=(𝔤d𝔤,[,]) inn(\mathfrak{g}) = ( \mathfrak{g} \stackrel{d}{\to} \mathfrak{g}, [-,-])

whose

  • elements in degree -1 are the elements x𝔤x \in \mathfrak{g}, thought of as inner degree-(-1) derivations

    ι x:CE(𝔤)CE(𝔤)\iota_x : CE(\mathfrak{g}) \to CE(\mathfrak{g})

    given by contraction with xx;

  • elements in degree 0 are the derivations of degree 0 that are of the form

    X:=[d CE(𝔤),ι x]:CE(𝔤)CE(𝔤) \mathcal{L}_X := [d_{CE(\mathfrak{g})}, \iota_x] : CE(\mathfrak{g}) \to CE(\mathfrak{g});

  • the differential d=[d CE,]:𝔤𝔤d = [d_{CE}, -] : \mathfrak{g} \to \mathfrak{g} is the commutator of derivations with the differential d CE(𝔤)d_{CE(\mathfrak{g})};

  • the bracket is the graded commutator of derivations.

Equivalently this is identified with the differential crossed module (𝔤Id𝔤)(\mathfrak{g} \stackrel{Id}{\to} \mathfrak{g}) with the action being the adjoint action of 𝔤\mathfrak{g} on itself.

One checks that for all x,y𝔤x, y \in \mathfrak{g} we have in inn(𝔤)inn(\mathfrak{g}) the brackets

  • [ι x,ι y]=0[\iota_x, \iota_y] = 0

  • [ x,ι y]=ι [x,y][\mathcal{L}_x, \iota_y] = \iota_{[x,y]}

  • [ x, y]= [x,y][\mathcal{L}_x, \mathcal{L}_y] = \mathcal{L}_{[x,y]}

and of course

  • x=[d,ι x] \mathcal{L}_x = [d, \iota_x] .

These identities are known as Cartan calculus. In this context x\mathcal{L}_x is called a Lie derivative.

In this sense one may understand inn(𝔤)inn(\mathfrak{g}) for general \infty-Lie algebras 𝔤\mathfrak{g} as providing an \infty-version of Cartan calculus.

Relation to other concepts

\infty-Lie algebra valued differential forms

For 𝔤\mathfrak{g} an ∞-Lie algebra, XX a smooth manifold, an ∞-Lie algebra valued differential form is a morphism

Ω (X)W(𝔤):A \Omega^\bullet(X) \leftarrow W(\mathfrak{g}) : A

of dg-algebras, from the Weil algebra into the de Rham complex of XX.

The image of the unshifted generators A: 1𝔤 *Ω (X)A : \wedge^1 \mathfrak{g}^* \to \Omega^\bullet(X) are the forms themselves, the image of the shifted generators F A: 1𝔤 *[1]F_A : \wedge^1 \mathfrak{g}^*[1] are the corresponding curvatures. The respect for the differential on the shifted generators are the Bianchi identity on the curvatures.

Precisely if the curvatures vanish does the morphism factor through the Chevalley-Eilenberg algebra W(𝔤)CE(𝔤)W(\mathfrak{g}) \to CE(\mathfrak{g}).

(F A=0)( CE(𝔤) A flat Ω (X) A W(𝔤)). (F_A = 0) \;\;\Leftrightarrow \;\; \left( \array{ && CE(\mathfrak{g}) \\ & {}^{\mathllap{\exists A_{flat}}}\swarrow & \uparrow \\ \Omega^\bullet(X) &\stackrel{A}{\leftarrow}& W(\mathfrak{g}) } \right) \,.

Invariant polynomials and Chern-Simons elements

A cocycle in the ∞-Lie algebra cohomology of the ∞-Lie algebra 𝔤\mathfrak{g} is a closed element in the Chevalley-Eilenberg algebra CE(𝔤)CE(\mathfrak{g}).

An invariant polynomial \langle -\rangle on 𝔤\mathfrak{g} is a closed element in the Weil algebra W(𝔤)\langle -\rangle \in W(\mathfrak{g}), subject to the additional condition that it its entirely in the shifted copy of 𝔤\mathfrak{g}, (𝔤 *[1])\langle - \rangle \in \wedge^\bullet (\mathfrak{g}^*[1]).

(𝔤 *[1]) \langle -\rangle \in \wedge^\bullet( \mathfrak{g}^*[1] )
d W(𝔤)=0. d_{W(\mathfrak{g})} \langle -\rangle = 0 \,.

For x𝔤x \in \mathfrak{g} an element of the \infty-Lie algebra, let

ι x:W(𝔤)W(𝔤) \iota_x : W(\mathfrak{g}) \to W(\mathfrak{g})

the evident operation of contraction with xx

ι x:tt(x) \iota_x : t \mapsto t(x)
ι x:σ(t)0 \iota_x : \sigma(t) \mapsto 0

extended as a graded derivation. Then the Lie derivative

x:=ad x:=[d W(𝔤),ι x]:W(𝔤)W(𝔤) \mathcal{L}_x := ad_x := [d_{W(\mathfrak{g})}, \iota_x] : W(\mathfrak{g}) \to W(\mathfrak{g})

encodes the coadjoint action of 𝔤\mathfrak{g} on 𝔤 *\mathfrak{g}^*. By the above definition of an invariant polynomial \langle - \rangle, we have

ι x=0 \iota_x \langle - \rangle = 0

and

d W(𝔤)=0 d_{W(\mathfrak{g})} \langle - \rangle = 0

and hence

ad x=0. ad_x \langle -\rangle = 0 \,.

Since the cohomology of W(𝔤)W(\mathfrak{g}) is trivial, there is necessarily for each invariant polynomial an element cs cs_{\langle -\rangle} such that

d W(𝔤)cs =. d_{W(\mathfrak{g})} cs_{\langle -\rangle} = \langle -\rangle \,.

This is the Chern-Simons element of the invariant polynomial. Notice, crucially, that this is ingeneral not restricted to the shifted part (𝔤 *[1])\wedge^\bullet (\mathfrak{g}^*[1]) Its restriction

μ :=cs | 𝔤 * \mu_{\langle -\rangle} := cs_{\langle - \rangle}|_{\wedge^\bullet \mathfrak{g}^*}

to the unshifted copy, hence to the Chevalley-Eilenberg algebra, is the cocycle that is in transgression with \langle - \rangle.

For

(A,F A):W(𝔤)Ω (X) (A,F_A) : W(\mathfrak{g}) \to \Omega^\bullet(X)

a collection of 𝔤\mathfrak{g}-valued differential forms (as above) and :CE(b n1)W(𝔤)\langle -\rangle : CE(b^{n-1}\mathbb{R}) \to W(\mathfrak{g}) an invariant polynomial, the composite

F A:CE(b n1)W(𝔤)(A,F A)Ω (X) \langle F_A\rangle : CE(b^{n-1}\mathbb{R}) \stackrel{\langle - \rangle}{\to} W(\mathfrak{g}) \stackrel{(A,F_A)}{\to} \Omega^\bullet(X)

is the corresponding curvature characteristic form, a closed nn-form on XX. For (,cs):W(b n1)W(𝔤)(\langle - \rangle, cs) : W(b^{n-1}) \to W(\mathfrak{g}) the corresponding Chern-Simons element we have that cs(A,F A)cs(A,F_A) is the corresponding Chern-Simons form on XX.

Examples

Weil algebra of a Lie algebra

Let 𝔤\mathfrak{g} be a finite dimensional Lie algebra. This Lie algebra regarded as a Lie algebroid has as base manifold the point, X 0=ptX_0 = pt. Its algebra of functions is accordingly the ground field, and the algebra C (X 0) 𝔤 *\wedge^\bullet_{C^\infty(X_0)} \mathfrak{g}^* is just a Grassmann algebra.

The Chevalley-Eilenberg algebra is

CE(𝔤)=( 𝔤 *,d 𝔤), CE(\mathfrak{g}) = (\wedge^\bullet \mathfrak{g}^*, d_{\mathfrak{g}}) \,,

where the differential acts on the elements of 𝔤 *\mathfrak{g}^* in degree 1 by the linear dual of the Lie bracket.

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

The corresponding Weil algebra is obtained by adding another copy of 𝔤 *\mathfrak{g}^* in degree 2

W(𝔤)=( (𝔤 *𝔤 *[1]),d W(𝔤)) W(\mathfrak{g}) = (\wedge^\bullet (\mathfrak{g}^* \oplus \mathfrak{g}^*[1]), d_{W(\mathfrak{g})})

where with σ:𝔤 *𝔤 *[1]\sigma : \mathfrak{g}^* \to \mathfrak{g}^*[1] the degree shift isomorphism, the differential acts as

d W(𝔤)| 𝔤 *:[,] *+σ d_{W(\mathfrak{g})}|_{\mathfrak{g}^*} : [-,-]^* + \sigma
d W(𝔤)| 𝔤 *[1]:σd CE(𝔤)σ 1. d_{W(\mathfrak{g})}|_{\mathfrak{g}^*[1]} : \sigma \circ d_{CE(\mathfrak{g})} \circ \sigma^{-1} \,.

For illustration, we spell this out in a basis.

Let {t a} a\{t_a\}_a be a basis for the underlying vector space of 𝔤\mathfrak{g} and let {C a bc}\{C^a{}_{b c}\} be the corresponding structure constants of the Lie bracket

[t b,t c]=C a bct a. [t_b, t_c] = C^a{}_{b c} t_a \,.

Then the Chevalley-Eilenberg algebra is generated on generators {t a}\{t^a\} of degree 1, on which the differential acts as

d CE(𝔤):t a12C a bct bt c. d_{CE(\mathfrak{g})} : t^a \mapsto - \frac{1}{2} C^a{}_{b c} t^b \wedge t^c \,.

The Weil algebra in turn is generated from these generators {t a}\{t^a\} in degree 1 and generators {r a}\{r^a\} in degree 2, with differential given by

d W(𝔤):t a12C a bct bt c+r a d_{W(\mathfrak{g})} : t^a \mapsto - \frac{1}{2} C^a{}_{b c} t^b \wedge t^c + r^a
d W(𝔤):r aC a bct br c. d_{W(\mathfrak{g})} : r^a \mapsto C^a{}_{b c} t^b \wedge r^c \,.

Weil algebra of a 0-Lie algebroid

A 0-truncated Lie algebroid is one for which the chain complex of modules over the TT-algebra in degree 0 vanishes:

CE(𝔞)=( A (𝔞 *),d CE(𝔞))=(A,d=0). CE(\mathfrak{a}) = (\wedge^\bullet_A (\mathfrak{a}^*), d_{CE(\mathfrak{a})}) = (A, d = 0) \,.

For instance for TT=CartSp the theory of smooth algebras, any smooth manifold XX regarded as an L-∞ algebroid is a 0-Lie algebroid with CE(X)=C (X)CE(X) = C^\infty(X) the smooth algebra of smooth functions on XX.

Observation

The Weil algebra of a 0-Lie algebroid XX is the Kähler de Rham complex of A=CE(X)A = CE(X):

W(𝔞)=(Ω (A),d dR). W(\mathfrak{a}) = (\Omega^\bullet(A), d_{dR}) \,.

This Weil algebra is the Chevalley-Eilenberg algebra of the tangent Lie algebroid TXT X of XX, which is the de Rham algebra Ω (X)\Omega^\bullet(X) of XX:

W(X)=CE(TX)=(Ω (X),d dR). W(X) = CE(T X) = (\Omega^\bullet(X), d_{dR}) \,.

References

Among the original references on Weil algebras for ordinary Lie algebras is

  • Henri Cartan, Cohomologie réelle d’un espace fibré principal diffrentielle I, II, Séminaire Henri Cartan, 1949/50, pp. 19-01 – 19-10 and 20-01 – 20-11, CBRM, (1950).

and

  • Henri Cartan, Notions d’algébre différentielle; application aux groupes de Lie et aux variétés ou opère un groupe de Lie , Colloque de topologie (espaces fibrs), Bruxelles, (1950), pp. 15–27.

This also explains the use of the Weil algebra in the calculation of the equivariant de Rham cohomology of manifolds acted on by a compact group. These papers are reprinted, explained and put in a modern context in the book

A clasical textbook account of standard material is in chapter VI, vol III of

Some remarks on the notation there as compared to ours: our d Wd_W is their δ W\delta_W on p. 226 (vol III). Their δ E\delta_E is our d CEd_{CE}. Their δ θ\delta_\theta is our d ρd_\rho (θ\theta/ρ\rho denoting the representation)..

Some related material is also in

  • Victor Guillemin, Shlomo Sternberg, Supersymmetry and equivariant de Rham theory, Springer, (1999).

The (obvious but conceptually important) observation that Lie algebra-valued 1-forms regarded as morphisms of graded vector spaces Ω (X) 1𝔤 *:A\Omega^\bullet(X) \leftarrow \wedge^1 \mathfrak{g}^* : A are equivalently morphisms of dg-algebras out of the Weil algebra Ω (X)W(𝔤):A\Omega^\bullet(X) \leftarrow W(\mathfrak{g}) : A and that one may think of as the identity W(𝔤)W(𝔤):IdW(\mathfrak{g}) \leftarrow W(\mathfrak{g}) : Id as the universal 𝔤\mathfrak{g}-connection appears in early articles for instance highlighted on p. 15 of

  • Franz W. Kamber; Philippe Tondeur, Semisimplicial Weil algebras and characteristic classes for foliated bundles in Cech cohomology , Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Stanford Univ., Stanford, Calif., 1973), Part 1, pp. 283–294. Amer. Math. Soc., Providence, R.I., (1975).

A survey of Weil algebras for Lie algebras is also available at

Weil algebra for L-infinity algebras and their role in defining invariant polynomials and Chern-Simons elements on \infty-Lie algebras from L-infinity algebra cocycle are considered in

Revised on November 3, 2013 05:12:33 by Urs Schreiber (82.113.106.128)