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.
The notion of Weil algebra is ordinarily defined for a Lie algebra $\mathfrak{g}$. It may be understood as the Chevalley-Eilenberg algebra of the tangent Lie 2-algebra $T \mathfrak{g}$ or $inn(\mathfrak{g})$ of $\mathfrak{g}$, generalizing the notion of tangent Lie algebroid $T X$ from a 0-truncated Lie algebroid $X$ (a smooth manifold) to the one-obeject Lie algebroid $\mathfrak{g}$.
Generally, for every Lie-∞-algebroid $\mathfrak{a}$ one may define the corresponding tangent Lie-$\infty$-algebroid $T \mathfrak{a}$, whose Chevalley-Eilenberg algebra may be called the Weil algebra of $\mathfrak{a}$:
Let $\mathfrak{g}$ be a finite-dimensional Lie algebra. The Weil algebra $W(\mathfrak{g})$ of $\mathfrak{g}$ is
the graded Grassmann algebra generated from the dual vector space $\mathfrak{g}^*$ together with another copy of $\mathfrak{g}^*$ shifted in degree
equipped with a derivation $d : W(\mathfrak{g}) \to W(\mathfrak{g})$ that makes this a dg-algebra, defined by the fact that on $\mathfrak{g}^*$ it acts as the differential of the Chevalley-Eilenberg algebra of $\mathfrak{g}$ plus the degree shift morphism $\mathfrak{g}^* \to \mathfrak{g}^*$.
This Weil algebra has trivial cohomology everywhere (except in degree 0 of course) and sits in a sequence
with the Chevalley-Eilenberg algebra of $\mathfrak{g}$ and its algebra of invariant polynomials on $\mathfrak{g}$. This may be understood as a model for the sequence of algebras of differential forms on the universal G-bundle
As such, the Weil algebra plays a crucial role in the study of the Lie algebra cohomology of $\mathfrak{g}$.
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.
Let $\mathfrak{g}$ be an L-∞ algebra of finite type. By our grading conventions this means that the graded vector space $\mathfrak{g}^*$ obtained by degreewise dualization is in non-negative degree, and $\wedge^1 \mathfrak{g}^* = \mathfrak{g}^*[1]$ is its shift up into positive degree.
A quick abstract way to characterize the Weil algebra of $\mathfrak{g}$ is as follows. Notice that there is a free functor/forgetful functor adjunction
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 .
The Weil algebra $W(\mathfrak{g})$ is the unique representative of the free dg-algebra on $\wedge^1 \mathfrak{g}^*$ for which the projection of graded vector spaces $\wedge^1(\mathfrak{g}^* \oplus \mathfrak{g}^*[1]) \to \wedge^1 \mathfrak{g}^*$ extended to a dg-algebra homomorphism $W(\mathfrak{g}) \to CE(\mathfrak{g})$
We discuss below in the Properties section that this is equivalent to the following component-wise definition
The Weil algebra $W(\mathfrak{g})$ is the semi-free dga whose underlying graded-commutative algebra is the exterior algebra
on $\mathfrak{g}^*$ and a shifted copy of $\mathfrak{g}^*$, and whose differential is the sum
of two graded derivations of degree +1 defined by
$\mathbf{d}$ acts by degree shift $\mathfrak{g}^* \to \mathfrak{g}^*[1]$ on elements in $\mathfrak{g}^*$ and by 0 on elements of $\mathfrak{g}^*[1]$;
$d_{CE(\mathfrak{g})}$ acts on unshifted elements in $\mathfrak{g}^*$ as the differential of the Chevalley-Eilenberg algebra of $\mathfrak{g}$ and is extended uniquely to shifted generators by graded-commutattivity
with $\mathbf{d}$:
for all $\omega \in \wedge^1 \mathfrak{g}^*$.
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 ($C^\infty$-ring).
So let $T$ be a Fermat theory. Write $T Alg$ for the corresponding category of algebra. There is a free functor/forgetful functor adjunction
to the category CRing of commutative Rings.
We need the facts that
a module over a $T$-algebra $A$ is uniquely specified by its underlying module over $U(A)$;
the universal derivation on a $T$-algebra $A$ is the de Rham differential
with values in the $A$-module of $T$-Kähler differentials.
See the corresponding entries for more details. The second point means that for $v : A \to N$ any $T$-derivation on $A$, there is a unique $A$-module homomorphism
such that the diagram
commutes.
Let now $\mathfrak{a}$ be an L-∞ algebroid with Chevalley-Eilenberg algebra considered as the following data;
a graded commutative semifree dga $CE(\mathfrak{a})$ over the ground field;
the structure of a $T$-algebra on the associative algebra $A := CE(\mathfrak{a})_0$ (over the ground field)
such that $d_{CE(\mathfrak{a})} : CE(\mathfrak{a})_0 \to CE(\mathfrak{a})_1$ is a derivation of $T$-algebra modules.
By semi-freeness there exists a $\mathbb{N}$-graded vector space $(\mathfrak{a}^*)^\bullet$ and an isomorphism
The Weil algebra $W(\mathfrak{a})$ of the $L_\infty$-algebroid $\mathfrak{a}$ is the Chevalley-Eilenberg algebra of the $L_\infty$-algebroid defined as follows
the $T$-algebra $A$ in degree 0 is the same as that of $\mathfrak{A}$;
the underlying graded algebra is the exterior algebra on $\mathfrak{a}^*$ and a shifted copy $\mathfrak{a}^*[1]$ as well as one copy of the Kähler differential module $\Omega^1$ in lowest degree (though of as the shifted copy of $A$ itself)
the differential is the sum
of two degree +1 graded derivations, where $d_{CE(\mathfrak{a})}$ and $\mathbf{a}$ are defined on $\wedge^1 \mathfrak{a}^* \oplus \mathfrak{a}^*[1]$ as above for $L_\infty$-algebras and on $A$ itself $d_{CE(\mathfrak{a})}$ vanishes and $\mathbf{d}$ acts as the universal derivation
The main point of the definition is that the differential restricted to the original (unshifted) generators is the original differential plus the shift:
By solving the condition $d_{W(\mathfrak{a})} \circ d_{W(\mathfrak{a})} = 0$ and using that $d_{CE(\mathfrak{a})} d_{CE(\mathfrak{a})} = 0$ this already fixes uniquely the differential $d_{W(\mathfrak{a})}$. To see this we only need to show that the value of $d_{W(\mathfrak{a})}(x)$ on a generator $x=\sigma(t) \in \mathfrak{a}^*[1]$ is completely determined by $d_{W(\mathfrak{a})}\vert_{\wedge^\bullet\mathfrak{a}^*}$. One computes:
and hence
This implies the following universal freeness property:
Let $\mathfrak{g}$ be an $L_\infty$-algebra. Morphisms of $dg$-algebras $W(\mathfrak{g}) \to A$ are in natural bijection to morphisms of graded vector spaces $\mathfrak{g}^* \to A$.
Forgetting the differential, $W(\mathfrak{g})$ is the free graded-commutative algebra generated by (a shifted copy of) $\mathfrak{g}^*$ and $\mathfrak{g}^*[1]$. Therefore,
Projecting down to $Hom_{grVect}(\mathfrak{g}^*,A)$, one obtains a natural map
which is a bijection.
To prove injectivity, we just have to show that the restriction of a dgca morphism $f:W(\mathfrak{g})\to A$ to $\mathfrak{g}^*$ determines the restriction of $f$ to $\mathfrak{g}^*[1]$. One has, for any $x=\sigma(t)\in \mathfrak{g}^*[1]$,
Since $d_{CE(\mathfrak{g})}(t)$ lies in the sub-gca of $W(\mathfrak{g})$ generated by $\mathfrak{g}^*$, the element $f(d_{CE(\mathfrak{g})}(t))$, and therefore $f(x)$, is determined by $f\vert_{\mathfrak{g}^*}$.
Next we show surjectivity, i.e. that every morphism of graded vector spaces $\phi:\mathfrak{g}^*\to A$ can be extended to a dgca morphism $f:W(\mathfrak{g})\to A$. Denote by $f_0: \wedge^\bullet \mathfrak{g}^*\to A$ the extension of $\phi$ to a graded commutative algebra morphism, and let $\psi:\mathfrak{g}^*[1]\to A$ be the graded vector space morphism defined by
for any $x=\sigma(t)\in \mathfrak{g}^*[1]$. The graded vector space morphism $\phi+\psi:\mathfrak{g}^*\oplus\mathfrak{g}^*[1]\to A$ extends to a commutative graded algebra $f:W(\mathfrak{g})\to A$, whose restriction to $\mathfrak{g}^*$ is $\phi$. We want to show that $f$ is actually a dgca morphism. We only need to test commutativity with the differentials on generators $t\in \mathfrak{g}^*$ and $x=\sigma(t)\in \mathfrak{g}^*[1]$. We have
which in particular implies that $d_A f\vert_{\wedge^\bullet \mathfrak{g}^*}=f d_{W(\mathfrak{g})}\vert_{\wedge^\bullet \mathfrak{g}^*}$, and
Since $d_{CE(\mathfrak{g})}(t)\in \wedge^\bullet \mathfrak{g}^*$, we obtain
For $A=CE(\mathfrak{g})$ the Chevalley-Eilenberg algebra of $\mathfrak{g}$, the inclusion $\mathfrak{g}^*\hookrightarrow CE(\mathfrak{g})$ induces a canonical surjective dgca morphism $W(\mathfrak{g})\to CE(\mathfrak{g})$. This is the identity on the unshifted generators, and 0 on the shifted generators.
For $A = \Omega^\bullet(X)$ the de Rham complex of a smooth manifold $X$, we have that
is the collection of total degree 1 differential forms with values in the $\infty$-Lie algebra $\mathfrak{g}$.
A morphism of
sends the unshifted generators $t^a$ to differential forms $A^a$, which one thinks of as local connection forms, and sends the shifted generators $\sigma t^a$ to their curvature. The respect for the differential on the shifted generators is the Bianchi identity on these curvatures.
A morphism $W(\mathfrak{g}) \to \Omega^\bullet(X)$ encodes a collection of flat $L_\infty$-algebra valued forms precisely if it factors by the canonical morphism $W(\mathfrak{g}) \to CE(\mathfrak{g})$ from above through the Chevalley-Eilenberg algebra of $\mathfrak{g}$.
The freeness property of the Weil algebra can be made more explicit by exhibiting a concrete isomorphism to the free dg-algebra on $\mathfrak{g}^*$.
The canonical free dg-algebra on $\mathfrak{g}^*$ is
where the differential $d_f$ is on the unshifted generators $t \in \mathfrak{g}^*$ the shift isomorphism $\sigma : \mathfrak{g}^* \to \mathfrak{g}^*[1]$ extended as a derivation and vanishes on the shifted generators
Or in other words, if $\bar \mathfrak{g}$ is the $\infty$-Lie algebra whose underlying graded vector space is that of $\mathfrak{g}$, but all whose brackets vanish, then
Notice the evident
The cochain cohomology of $F(\mathfrak{g})$ vanishes in positive degree.
To see this, let $K := \sigma^{-1} : F(\mathfrak{g}) \to F(\mathfrak{g})$ be the degree down-shift isomorphism $\mathfrak{g}^*[1] \to \mathfrak{g}^*$ extended as a graded derivation of degree -1, then
and hence for any $\omega \in F(\mathfrak{g})$ such that $d_{F(\mathfrak{g})} \omega = 0$ we have $\omega = d_{F(\mathfrak{g})} K \omega$.
Given $\mathfrak{g}$, there is an isomorphism of dg-algebras
given by
It is clear that $f$ is a dg-algebra homomorphism. The inverse dg-algebra morphism is given on generators by
Note that the isomorphism $f$ is precisely the dgca isomorphism induced between $W(\overline\mathfrak{g})$ and $W(\mathfrak{g})$ by the identity of $\mathfrak{g}^*$ as a graded vector spaces morphism $\overline{\mathfrak{g}}^*\to\mathfrak{g}^*$.
The cochain cohomology of the Weil algebra of an $L_\infty$-algebra is trivial.
This means that homotopy-theoretically the Weil algebra is the point. Dually, the $\infty$-Lie algebra $inn(\mathfrak{g})$ is a model for the point. In fact, one can see that $inn(\mathfrak{g})$ is the universal principal ∞-bundle over $\mathfrak{g}$ 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.
The Weil algebra may be identified with the Cartan model for equivariant de Rham cohomology for the special case of the Lie group $G$ 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.
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 $W(\mathfrak{g})$ of some L-∞ algebra $\mathfrak{g}$ is the Chevalley-Eilenberg algebra of another $\infty$-Lie algebra.
For any $\infty$-Lie algebra $\mathfrak{g}$ write $inn(\mathfrak{g})$ for the $\infty$-Lie algebra whose CE-algebra is $W(\mathfrak{g})$:
In the following we discuss these inner automorphism $\infty$-Lie algebras in more detail. (See section 6 of (SSSI)).
For $\mathfrak{g}$ an ordinary Lie algebra the inner derivation Lie 2-algebra is the strict Lie 2-algebra given by the dg-Lie algebra
whose
elements in degree -1 are the elements $x \in \mathfrak{g}$, thought of as inner degree-(-1) derivations
$\iota_x : CE(\mathfrak{g}) \to CE(\mathfrak{g})$
given by contraction with $x$;
elements in degree 0 are the derivations of degree 0 that are of the form
$\mathcal{L}_X := [d_{CE(\mathfrak{g})}, \iota_x] : CE(\mathfrak{g}) \to CE(\mathfrak{g})$;
the differential $d = [d_{CE}, -] : \mathfrak{g} \to \mathfrak{g}$ is the commutator of derivations with the differential $d_{CE(\mathfrak{g})}$;
the bracket is the graded commutator of derivations.
Equivalently this is identified with the differential crossed module $(\mathfrak{g} \stackrel{Id}{\to} \mathfrak{g})$ with the action being the adjoint action of $\mathfrak{g}$ on itself.
One checks that for all $x, y \in \mathfrak{g}$ we have in $inn(\mathfrak{g})$ the brackets
$[\iota_x, \iota_y] = 0$
$[\mathcal{L}_x, \iota_y] = \iota_{[x,y]}$
$[\mathcal{L}_x, \mathcal{L}_y] = \mathcal{L}_{[x,y]}$
and of course
These identities are known as Cartan calculus. In this context $\mathcal{L}_x$ is called a Lie derivative.
In this sense one may understand $inn(\mathfrak{g})$ for general $\infty$-Lie algebras $\mathfrak{g}$ as providing an $\infty$-version of Cartan calculus.
For $\mathfrak{g}$ an ∞-Lie algebra, $X$ a smooth manifold, an ∞-Lie algebra valued differential form is a morphism
of dg-algebras, from the Weil algebra into the de Rham complex of $X$.
The image of the unshifted generators $A : \wedge^1 \mathfrak{g}^* \to \Omega^\bullet(X)$ are the forms themselves, the image of the shifted generators $F_A : \wedge^1 \mathfrak{g}^*[1]$ 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 $W(\mathfrak{g}) \to CE(\mathfrak{g})$.
A cocycle in the ∞-Lie algebra cohomology of the ∞-Lie algebra $\mathfrak{g}$ is a closed element in the Chevalley-Eilenberg algebra $CE(\mathfrak{g})$.
An invariant polynomial $\langle -\rangle$ on $\mathfrak{g}$ is a closed element in the Weil algebra $\langle -\rangle \in W(\mathfrak{g})$, subject to the additional condition that it its entirely in the shifted copy of $\mathfrak{g}$, $\langle - \rangle \in \wedge^\bullet (\mathfrak{g}^*[1])$.
For $x \in \mathfrak{g}$ an element of the $\infty$-Lie algebra, let
the evident operation of contraction with $x$
extended as a graded derivation. Then the Lie derivative
encodes the coadjoint action of $\mathfrak{g}$ on $\mathfrak{g}^*$. By the above definition of an invariant polynomial $\langle - \rangle$, we have
and
and hence
Since the cohomology of $W(\mathfrak{g})$ is trivial, there is necessarily for each invariant polynomial an element $cs_{\langle -\rangle}$ such that
This is the Chern-Simons element of the invariant polynomial. Notice, crucially, that this is ingeneral not restricted to the shifted part $\wedge^\bullet (\mathfrak{g}^*[1])$ Its restriction
to the unshifted copy, hence to the Chevalley-Eilenberg algebra, is the cocycle that is in transgression with $\langle - \rangle$.
For
a collection of $\mathfrak{g}$-valued differential forms (as above) and $\langle -\rangle : CE(b^{n-1}\mathbb{R}) \to W(\mathfrak{g})$ an invariant polynomial, the composite
is the corresponding curvature characteristic form, a closed $n$-form on $X$. For $(\langle - \rangle, cs) : W(b^{n-1}) \to W(\mathfrak{g})$ the corresponding Chern-Simons element we have that $cs(A,F_A)$ is the corresponding Chern-Simons form on $X$.
Let $\mathfrak{g}$ be a finite dimensional Lie algebra. This Lie algebra regarded as a Lie algebroid has as base manifold the point, $X_0 = pt$. Its algebra of functions is accordingly the ground field, and the algebra $\wedge^\bullet_{C^\infty(X_0)} \mathfrak{g}^*$ is just a Grassmann algebra.
The Chevalley-Eilenberg algebra is
where the differential acts on the elements of $\mathfrak{g}^*$ in degree 1 by the linear dual of the Lie bracket.
The corresponding Weil algebra is obtained by adding another copy of $\mathfrak{g}^*$ in degree 2
where with $\sigma : \mathfrak{g}^* \to \mathfrak{g}^*[1]$ the degree shift isomorphism, the differential acts as
For illustration, we spell this out in a basis.
Let $\{t_a\}_a$ be a basis for the underlying vector space of $\mathfrak{g}$ and let $\{C^a{}_{b c}\}$ be the corresponding structure constants of the Lie bracket
Then the Chevalley-Eilenberg algebra is generated on generators $\{t^a\}$ of degree 1, on which the differential acts as
The Weil algebra in turn is generated from these generators $\{t^a\}$ in degree 1 and generators $\{r^a\}$ in degree 2, with differential given by
A 0-truncated Lie algebroid is one for which the chain complex of modules over the $T$-algebra in degree 0 vanishes:
For instance for $T$=CartSp the theory of smooth algebras, any smooth manifold $X$ regarded as an L-∞ algebroid is a 0-Lie algebroid with $CE(X) = C^\infty(X)$ the smooth algebra of smooth functions on $X$.
The Weil algebra of a 0-Lie algebroid $X$ is the Kähler de Rham complex of $A = CE(X)$:
This Weil algebra is the Chevalley-Eilenberg algebra of the tangent Lie algebroid $T X$ of $X$, which is the de Rham algebra $\Omega^\bullet(X)$ of $X$:
Among the original references on Weil algebras for ordinary Lie algebras is
and
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 $d_W$ is their $\delta_W$ on p. 226 (vol III). Their $\delta_E$ is our $d_{CE}$. Their $\delta_\theta$ is our $d_\rho$ ($\theta$/$\rho$ denoting the representation)..
Some related material is also in
The (obvious but conceptually important) observation that Lie algebra-valued 1-forms regarded as morphisms of graded vector spaces $\Omega^\bullet(X) \leftarrow \wedge^1 \mathfrak{g}^* : A$ are equivalently morphisms of dg-algebras out of the Weil algebra $\Omega^\bullet(X) \leftarrow W(\mathfrak{g}) : A$ and that one may think of as the identity $W(\mathfrak{g}) \leftarrow W(\mathfrak{g}) : Id$ as the universal $\mathfrak{g}$-connection appears in early articles for instance highlighted on p. 15 of
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