Weil algebra in nLab

Contents

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/local Artin algebra.

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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-object 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}) \coloneqq \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

G→ℰG→ℬG. 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

Plain Weil algebra

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

(F⊣U):dgAlg⟶U←FVect[ℤ] (F \dashv U) \colon dgAlg \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\longrightarrow}} Vect[\mathbb{Z}]

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

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

Adjusted Weil algebras

For some purposes of L ∞ L_\infty -algebra valued connection, the above definition of Weil algebra of an L ∞L_\infty-algebra is not quite appropriate. While Def. gives the Weil algebra up to compatible isomorphism, in applications there is in fact the freedom to choose it up to compatible quasi-isomorphism, and this freedom allows to find better representatives.

For more on this see at adjusted Weil algebra.

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

(F⊣U):TAlg→U←FCRing (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:A→Ω 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:A→Nv : 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 \coloneqq CE(\mathfrak{a})_0 (over the ground field)

   such that d CE(𝔞):CE(𝔞) 0→CE(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 ℕ\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})}) \,.

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:

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}^*.

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

To see this, let K≔σ −1:F(𝔤)→F(𝔤)K \coloneqq \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 ω∈F(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.

Characterization in the smooth ∞\infty-topos

The Weil algebra of a Lie algebra is naturally identified with the de Rham algebra of differential forms on the “universal GG-principal bundle with connection” in its stacky incarnation (Freed-Hopkins 13):

Write BG conn≃Ω(−,𝔤)//G\mathbf{B}G_{conn}\simeq \mathbf{\Omega}(-,\mathfrak{g})//G for the universal moduli stack of GG-principal connections (as discussed there), a smooth groupoid. The quotient projection may be regarded as the universal GG-connection:

Ω flat(−,𝔤) ↓ EG conn ≔ Ω(−,𝔤) ↓ ↓ BG conn ≔ Ω(−,𝔤)//G \array{ && \mathbf{\Omega}_{flat}(-,\mathfrak{g}) \\ && \downarrow \\ \mathbf{E}G_{conn} &\coloneqq & \mathbf{\Omega}(-,\mathfrak{g}) \\ \downarrow && \downarrow \\ \mathbf{B}G_{conn} &\coloneqq &\mathbf{\Omega}(-,\mathfrak{g})//G }

(After forgetting the connection/form data this is just the universal principal bundle EG→BG\mathbf{E}G \to \mathbf{B}G)

The differential kk-forms on a smooth groupoid XX are just homs X→Ω k(−)X \to \mathbf{\Omega}^k(-) into the sheaf of kk-forms. (See at geometry of physics – differential forms). These Ω k(X)\Omega^k(X) inherit the de Rham differential and hence form the de Rham complex of the stack. (Notice that this is very different from the hom of XX into a shift of the full de Rham complex regarded as a sheaf of complexes. The latter is instead a model for the real ordinary cohomology of XX, see at smooth infinity-groupoid – structures for more on this).

One finds (Freed-Hopkins 13) that the de Rham complex, in this sense, of EG conn\mathbf{E}G_{conn} is the Weil algebra:

Ω •(EG conn)≔Ω •(Ω(−,𝔤))≃W(𝔤). \Omega^\bullet(\mathbf{E}G_{conn}) \coloneqq \Omega^\bullet( \mathbf{\Omega}(-,\mathfrak{g}) ) \simeq W(\mathfrak{g}) \,.

Turning this around, this motivates to algebraically define the connection on a principal ∞-bundle, via Lie integration, as discussed there.

Relation to Cartan model for equivariant de Rham cohomology

The Weil algebra may be identified with the Cartan model for equivariant de Rham cohomology for the special case of the Lie group GG acting on itself by right multiplication. Concersely, the Cartan models form a generalization of the Weil algebra. See at equivariant de Rham cohomology – Cartan model for more.

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.

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

For an ordinary Lie algebra

Equivalently this is identified with the differential crossed module (𝔤→Id𝔤)(\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,y∈𝔤x, y \in \mathfrak{g} we have in inn(𝔤)inn(\mathfrak{g}) the brackets

• [ι x,ι y]=0[\iota_x, \iota_y] = 0

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

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

and of course

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

These identities are known as Cartan calculus. In this context ℒ 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 x∈𝔤x \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:t↦t(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

ℒ x≔ad x≔[d W(𝔤),ι x]:W(𝔤)→W(𝔤) \mathcal{L}_x \coloneqq ad_x \coloneqq [d_{W(\mathfrak{g})}, \iota_x] \colon 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} \coloneqq 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) \colon W(\mathfrak{g}) \to \Omega^\bullet(X)

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

⟨F A⟩:CE(b n−1ℝ)→⟨−⟩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 n−1)→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]:σ∘d 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 a↦−12C a bct b∧t 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 a↦−12C a bct b∧t 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 a↦C a bct b∧r 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.

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}) \,.

• ∞-Lie algebra cohomology

• Chevalley-Eilenberg algebra

• Weil algebra

• invariant polynomial

• Sullivan model of free loop space

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

In monographs:

• Werner Greub, Stephen Halperin, Ray Vanstone, chapter VI, vol III of: Connections, Curvature, and Cohomology Academic Press (1973)

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)..

In the context of equivariant de Rham cohomology:

• Michael Atiyah, Raoul Bott, The moment map and equivariant cohomology, Topology 23, 1 (1984) (doi:10.1016/0040-9383(84)90021-1, pdf)

• Jaap Kalkman, BRST model applied to symplectic geometry, Ph.D. Thesis, Utrecht, 1993 (arXiv:hep-th/9308132, cds:9308132, euclid:1104252784)

and with an eye towards supersymmetry:

• Mauri Miettinen, Weil Algebras and Supersymmetry (arXiv:hep-th/9612209, cds:317377, spire:427720)

• Victor Guillemin, Shlomo Sternberg, Supersymmetry and equivariant de Rham theory, Springer, (1999) (doi:10.1007/978-3-662-03992-2)

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

• Encyclopedia of Mathematics: Weil algebra of a Lie algebra

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

• Hisham Sati, Urs Schreiber, Jim Stasheff, L ∞ L_{\infty} algebra connections and applications to String- and Chern-Simons n n -transport, in Quantum Field Theory, Birkhäuser (2009) 303-424 [arXiv:0801.3480, doi:10.1007/978-3-7643-8736-5_17]

The abstract characterization is due to

• Daniel Freed, Michael Hopkins, Chern-Weil forms and abstract homotopy theory, Bull. Amer. Math. Soc. 50 (2013), 431-468 (arXiv:1301.5959)

Further discussion of Weil algebras for the string Lie 2-algebra:

• Lennart Schmidt, Twisted Weil Algebras for the String Lie 2-Algebra, in Christian Saemann, Urs Schreiber, Martin Wolf (eds.) Higher Structures in M-Theory, Durham Symposium 2018, Fortschritte der Physik 2019 (arXiv:1903.02873)