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/local Artin algebra.

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Idea

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-object 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 $\mathfrak{a}$:

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

Weil algebra of a Lie algebra

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

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

• 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

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

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

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

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

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 $L_\infty$-algebras

Plain Weil 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

$$(F \dashv U) \colon dgAlg \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\longrightarrow}} Vect[\mathbb{Z}]$$

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

Definition

(plain Weil algebra of $L_\infty$-algebra)
The Weil algebra $W(\mathfrak{g})$ of an $L_\infty$-algebra is the unique representative of the free dg-algebra on $\wedge^1 \mathfrak{g}^*$ for which the projection of graded vector spaces $\wedge^1(\mathfrak{g}^* \oplus \mathfrak{g}^*[1]) \to \wedge^1 \mathfrak{g}^*$ extends 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

Definition

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

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

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

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

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-commutativity

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

  with $\mathbf{d}$:

  $$d_{CE(\mathfrak{g})} \mathbf{d} \omega \coloneqq - \mathbf{d} d_{CE(\mathfrak{g})} \omega$$

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

Adjusted Weil algebras

For some purposes of $L_\infty$-algebra valued connection, the above definition of Weil algebra of an $L_\infty$-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 $L_\infty$-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 ($C^\infty$-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

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

to the category CRing of commutative Rings.

We need the facts that

• a module over a $T$-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

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

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

$$\Omega^\bullet(A) \to N$$

such that the diagram

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

commutes.

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

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

2. the structure of a $T$-algebra on the associative algebra $A \coloneqq CE(\mathfrak{a})_0$ (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

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

Definition

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)

  $$\wedge^\bullet (\Omega^1(A) \oplus (\mathfrak{a}^*) \oplus \mathfrak{a}^*[1]) \,.$$

• the differential is the sum

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

  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

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

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:

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

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:

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

and hence

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

This implies the following universal freeness property:

Proposition

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 $\mathfrak{g}^* \to A$.

Proof

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,

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

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

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

which is a bijection.

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

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

Since $d_{CE(\mathfrak{g})}(t)$ 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

$$\psi(x)=d_A \phi(t)-f_0d_{CE(\mathfrak{g})}(t),$$

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

$$d_A f(t)=d_A\phi(t)=\psi(\sigma(t))+f_0d_{CE(\mathfrak{g})}(t)=f(\sigma(t))+ f d_{CE(\mathfrak{g})}(t)=f d_{W(\mathfrak{g})}(t),$$

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

$$d_A f(x)= d_A \psi(x) = -d_A f_0d_{CE(\mathfrak{g})}(t)=-d_A f (d_{CE(\mathfrak{g})}(t)).$$

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

$$d_A f(x)= -f d_{W(\mathfrak{g})} (d_{CE(\mathfrak{g})}(t))= -f d_{W(\mathfrak{g})}(d_{W(\mathfrak{g})}(t)-x)=f d_{W(\mathfrak{g})}(x).$$

Example

For $A=CE(\mathfrak{g})$ 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.

Example

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

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

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

A morphism of

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

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 $\mathfrak{g}^*$.

Definition

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

$$F(\mathfrak{g}) \coloneqq \wedge^\bullet( \mathfrak{g}^* \oplus \mathfrak{g}^*[1], d_F )$$

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

$$d_F : t \mapsto \sigma(t) \,,$$

$$d_F : \sigma(t) \mapsto 0 \,.$$

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

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

Notice the evident

Observation

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

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

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

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

Lemma

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

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

given by

$$f : t \mapsto t$$

$$f : \sigma(t) \mapsto d_{W(\mathfrak{g})} t = d_{CE(\mathfrak{g})} t + \sigma(t) \,.$$

Proof

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

$$f^{-1} : t \mapsto t$$

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

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 $\overline{\mathfrak{g}}^*\to\mathfrak{g}^*$.

Corollary

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

Remark

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.

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 $G$-principal bundle with connection” in its stacky incarnation (Freed-Hopkins 13):

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

$$\array{ && \mathbf{\Omega}_{flat}(-,\mathfrak{g}) \\ && \downarrow \\ \mathbf{E}G_{conn} &\coloneqq & \mathbf{\Omega}(-,\mathfrak{g}) \\ \downarrow && \downarrow \\ \mathbf{B}G_{conn} &\coloneqq &\mathbf{\Omega}(-,\mathfrak{g})//G }$$

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

The differential $k$-forms on a smooth groupoid $X$ are just homs $X \to \mathbf{\Omega}^k(-)$ into the sheaf of $k$-forms. (See at geometry of physics – differential forms). These $\Omega^k(X)$ 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 $X$ 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 $X$, see at smooth infinity-groupoid – structures for more on this).

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

$$\Omega^\bullet(\mathbf{E}G_{conn}) \coloneqq \Omega^\bullet( \mathbf{\Omega}(-,\mathfrak{g}) ) \simeq W(\mathfrak{g}) \,.$$

Chevalley-Eilenberg algebra CE $\leftarrow$ Weil algebra W $\leftarrow$ invariant polynomials inv

differential forms on moduli stack $\mathbf{B}G_{conn}$ of principal connections (Freed-Hopkins 13):

$$\array{ CE(\mathfrak{g}) &\simeq& \Omega^\bullet_{li \atop cl}(G) \\ \uparrow && \uparrow \\ W(\mathfrak{g}) &\simeq & \Omega^\bullet(\mathbf{E}G_{conn}) & \simeq & \Omega^\bullet(\mathbf{\Omega}(-,\mathfrak{g})) \\ \uparrow && \uparrow \\ inv(\mathfrak{g}) &\simeq& \Omega^\bullet(\mathbf{B}G_{conn}) & \simeq & \Omega^\bullet(\mathbf{\Omega}(-,\mathfrak{g})/G) }$$

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 $G$ 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.

As the CE-algebra of the $L_\infty$-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 $W(\mathfrak{g})$ of some L-∞ algebra $\mathfrak{g}$ is the Chevalley-Eilenberg algebra of another $\infty$-Lie algebra.

Definition

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

$$CE(inn(\mathfrak{g})) \coloneqq W(\mathfrak{g}) \,.$$

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

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

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

  $\mathcal{L}_X \coloneqq [d_{CE(\mathfrak{g})}, \iota_x] \colon CE(\mathfrak{g}) \to CE(\mathfrak{g})$;

• 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

• $\mathcal{L}_x = [d, \iota_x]$.

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.

Relation to other concepts

$\infty$-Lie algebra valued differential forms

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

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

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 $W(\mathfrak{g}) \to CE(\mathfrak{g})$.

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

Invariant polynomials and Chern-Simons elements

A cocycle in the ∞-Lie algebra cohomology of the ∞-Lie algebra $\mathfrak{g}$ 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 $\mathfrak{g}$, $\langle - \rangle \in \wedge^\bullet (\mathfrak{g}^*[1])$.

$$\langle -\rangle \in \wedge^\bullet( \mathfrak{g}^*[1] )$$

$$d_{W(\mathfrak{g})} \langle -\rangle = 0 \,.$$

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

$$\iota_x : W(\mathfrak{g}) \to W(\mathfrak{g})$$

the evident operation of contraction with $x$

$$\iota_x : t \mapsto t(x)$$

$$\iota_x : \sigma(t) \mapsto 0$$

extended as a graded derivation. Then the Lie derivative

$$\mathcal{L}_x \coloneqq ad_x \coloneqq [d_{W(\mathfrak{g})}, \iota_x] \colon W(\mathfrak{g}) \to W(\mathfrak{g})$$

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

$$\iota_x \langle - \rangle = 0$$

and

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

and hence

$$ad_x \langle -\rangle = 0 \,.$$

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

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

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

$$\mu_{\langle -\rangle} \coloneqq cs_{\langle - \rangle}|_{\wedge^\bullet \mathfrak{g}^*}$$

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

For

$$(A,F_A) \colon W(\mathfrak{g}) \to \Omega^\bullet(X)$$

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

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

is the corresponding curvature characteristic form, a closed $n$-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 $X$.

Examples

Weil algebra of a Lie algebra

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

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

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

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

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

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

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

$$d_{W(\mathfrak{g})}|_{\mathfrak{g}^*} : [-,-]^* + \sigma$$

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

For illustration, we spell this out in a basis.

Let $\{t_a\}_a$ 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

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

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

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

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

$$d_{W(\mathfrak{g})} : t^a \mapsto - \frac{1}{2} C^a{}_{b c} t^b \wedge t^c + r^a$$

$$d_{W(\mathfrak{g})} : r^a \mapsto C^a{}_{b c} t^b \wedge r^c \,.$$

Weil algebra of a 0-Lie algebroid

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

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

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 $X$.

Observation

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

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

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 $X$:

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

Related concepts

• ∞-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 $d_W$ 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)..

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 $\Omega^\bullet(X) \leftarrow \wedge^1 \mathfrak{g}^* : A$ 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

• 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, $L_{\infty}$ algebra connections and applications to String- and Chern-Simons $n$-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)