nLab
Weil algebra

Idea

Ordinarily one speaks of the Weil algebra of a Lie algebra $𝔤$. We briefly recall this concept and then observe its abstract nonsense meaning from the nPOV as a mapping cone.

Weil algebra of a Lie algebra

The Weil algebra of a Lie algebra $𝔤$ is the

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

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

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

This Weil algebra has trivial cohomology and sits in a sequence

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

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

$$G\to ℰG\to ℬG.$$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 $𝔤$.

General interpretation as a mapping cone

We observe that the Weil dg-algebra $W(𝔤)$ defined this way is the mapping cone object of the identity morphism on the Chevalley-Eilenberg algebra $\mathrm{CE}(𝔤)$ in the model structure on dg-algebras ${\mathrm{dgAlg}}^{\mathrm{op}}$.

Another way to say this is: in as much as a Chevalley-Eilenberg algebra of a Lie $\mathrm{\infty }$-algebroid is the algebra of functions on an NQ-supermanifold $X$, the Weil algebra is the algebra of functions on the shifted tangent bundle $T[1]X$.

One can also understand this as generalizing the notion of superdifferential form from ordinary supermanifolds to NQ-supermanifolds.

Weil algebra of an $L_{\mathrm{\infty }}$-algebra

Since every L-infinity-algebra $𝔤$ is equally defined by its Chevalley-Eilenberg algebra, the interpretation of the Weil algebra as a mapping cone gives a definition of Weil algebras of $L_{\mathrm{\infty }}$-algebras. And of L-infinity-algebroids.

…

Definition

For $\mathrm{CE}(𝔤)={\wedge }_{C^{\mathrm{\infty }}(X_0)}^{•}{𝔤}^{*}$ the Chevalley-Eilenberg algebra of a $L_{\mathrm{\infty }}$-algebroid $g$ over a manifold $X_0$, the corresponding Weil algebra is

$$\mathrm{W}(𝔤):=({\wedge }_{C^{\mathrm{\infty }}(X_0)}^{•}({𝔤}^{*}\oplus \Gamma (T^{*}{X}_0)\oplus {𝔤}^{*}[1]),d_{\mathrm{W}(𝔤)})$$\mathrm{W}(\mathfrak{g}) := \left( \wedge^\bullet_{C^\infty(X_0)} ( \mathfrak{g}^* \oplus \Gamma(T^* X_0) \oplus \mathfrak{g}^*[1] ) , d_{\mathrm{W}(\mathfrak{g})} \right)

where the differential acts as the deRham differential on ${\wedge }^{•}\Gamma (T^{*}X)$ and is defined on generators in ${𝔤}^{*}\oplus {𝔤}^{*}[1]$ by

$$d_{\mathrm{W}(𝔤)}=(\begin{array}{cc}{d}_{\mathrm{CE}(𝔤)}& 0\\ \sigma & -\sigma \circ d_{\mathrm{CE}(𝔤)}\circ {\sigma }^{-1}\end{array},)$$d_{\mathrm{W}(\mathfrak{g})} = \left( \array{ d_{\mathrm{CE}(\mathfrak{g})} & 0 \\ \sigma & - \sigma \circ d_{\mathrm{CE}(\mathfrak{g})} \circ \sigma^{-1} } \,, \right)

where $\sigma {\mid }_{{𝔤}^{*}}:{𝔤}^{*}\to {𝔤}^{*}[1]$ is the canonical degree-shifting isomorphism.

Examples

Weil algebra of a Lie algebra

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

The Chevalley-Eilenberg algebra is

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

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

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

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

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

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

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

$$d_{W(𝔤)}{\mid }_{{𝔤}^{*}[1]}:\sigma \circ d_{\mathrm{CE}(𝔤)}\circ {\sigma }^{-1}.$$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 $\{t_a{\}}_a$ be a basis for the underlying vector space of $𝔤$ and let $\{C^a{}_{bc}\}$ be the corresponding structure constants of the Lie bracket

$$[t_b,t_c]=C^a{}_{bc}{t}_a.$$[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

$$d_{\mathrm{CE}(𝔤)}:t^a\mapsto -\frac{1}{2}{C}^a{}_{bc}{t}^b\wedge t^c.$$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 $\{t^a\}$ in degree 1 and generators $\{r^a\}$ in degree 2, with differential given by

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

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

References

The use of the Weil algebra in the calculation of the equivariant deRham cohomology of manifolds acted on by a compact group goes at least back to two papers by H. Cartan from 1950. These papers are reprinted, explained and put in a modern context in the book

• Victor Guillemin, Shlomo Sternberg, Supersymmetry and Equivariant de Rham Theory, Springer, 1999.

For the role played by the Weil algebra in the general context of higher Lie theory see

• Hisham Sati, Urs Schreiber, Jim Stasheff, $L_{\mathrm{\infty }}$ algebra connections and applications to String- and Chern-Simons $n$-transport (arXiv)