Schreiber
invariant polynomial on ∞-Lie algebroid

Definition

The dg-algebra of invariant polynmials on an ∞-Lie algebroid $𝔞$ is the sub-Chevalley-Eilenberg algebra

$$\mathrm{inv}(𝔞)\subset \mathrm{CE}(\Sigma 𝔞)$$inv(\mathfrak{a}) \subset CE(\Sigma\mathfrak{a})

of the suspension $\Sigma 𝔞$ of $𝔞$ that is universal with respect to the property that (with respect to the model below) it fits into diagrams

$$\begin{array}{ccc}\mathrm{CE}(𝔤)& \stackrel{\swarrow \Downarrow {\nwarrow }^0}{\nwarrow \swarrow }& \mathrm{CE}(𝔤)\\ \uparrow & & \uparrow \\ W(𝔤)& \stackrel{\swarrow \Downarrow {\nwarrow }^0}{\nwarrow \swarrow }& W(𝔤)\\ \uparrow & & \uparrow \\ \mathrm{inv}(𝔤)& \stackrel{0}{\leftarrow }& \mathrm{inv}(𝔤)\end{array}$$\array{ CE(\mathfrak{g}) &\stackrel{\swarrow \Downarrow\nwarrow^{0}}{\nwarrow\swarrow}& CE(\mathfrak{g}) \\ \uparrow && \uparrow \\ W(\mathfrak{g}) &\stackrel{\swarrow \Downarrow\nwarrow^{0}}{\nwarrow\swarrow}& W(\mathfrak{g}) \\ \uparrow && \uparrow \\ inv(\mathfrak{g}) &\stackrel{0}{\leftarrow}& inv(\mathfrak{g}) }

of graded derivations.

Model

Let $H=(\mathrm{SPSh}(C{)}^{\mathrm{loc}}{)}^{\circ }$ be the ambient smooth (∞,1)-topos.

To construct the algebra invariant polynomials on an ∞-Lie algebroid $𝔞$ is most convenient to use the injective local model structure on simplicial presheaves, $\mathrm{SPSh}(C{)}_{\mathrm{inj}}^{\mathrm{loc}}$.

Since in this structure every object is cofibrant, we may compute the homotopy pushout

$$\begin{array}{ccc}𝔞& \to & *\\ \downarrow & & \downarrow \\ *& \to & \Sigma 𝔞\end{array}$$\array{ \mathfrak{a} &\to& {*} \\ \downarrow && \downarrow \\ {*} &\to& \Sigma \mathfrak{a} }

that defines the suspension object $\Sigma 𝔞$ by the ordinary colimit diagram

$$\begin{array}{ccccc}& & 𝔞& \to & *\\ & & \downarrow & & \downarrow \\ 𝔞& \to & \mathrm{cyl}(𝔞)\\ \downarrow & & & & \downarrow \\ *& \to & & \to & \Sigma 𝔞\end{array}$$\array{ && \mathfrak{a} &\to& {*} \\ && \downarrow && \downarrow \\ \mathfrak{a} &\to& cyl(\mathfrak{a}) \\ \downarrow &&&& \downarrow \\ {*} &\to& &\to& \Sigma \mathfrak{a} }

in $\mathrm{SPSh}(C{)}_{\mathrm{inj}}^{\mathrm{loc}}$, where $\mathrm{cyl}(𝔞)$ is some cylinder object of $𝔞$, as described at mapping cone.

This in turn may be computed by two successive ordinary pushouts squares

$$\begin{array}{ccccc}& & 𝔞& \to & *\\ & & \downarrow & & \downarrow \\ 𝔞& \to & \mathrm{cyl}(𝔞)\\ \downarrow & & \downarrow & & \downarrow \\ *& \to & \mathrm{cone}(𝔞)& \to & \Sigma 𝔞\end{array}$$\array{ && \mathfrak{a} &\to& {*} \\ && \downarrow && \downarrow \\ \mathfrak{a} &\to& cyl(\mathfrak{a}) \\ \downarrow &&\downarrow&& \downarrow \\ {*} &\to& cone(\mathfrak{a}) &\to& \Sigma \mathfrak{a} }

where $\mathrm{cone}(𝔞)$ is the cone of $𝔞$.

The Chevalley-Eilenberg algebra-functor

$$\mathrm{CE}(-):\mathrm{SPSh}(C{)}_{\mathrm{inj}}^{\mathrm{loc}}\to {\mathrm{dgAlg}}^{\mathrm{op}}$$CE(-) : SPSh(C)_{inj}^{loc} \to dgAlg^{op}

is a left Quillen functor and hence preserves this colimit, sending it to the limit diagram in $\mathrm{dgAlg}$

$$\begin{array}{ccccc}& & \mathrm{CE}(𝔞)& \leftarrow & ℝ\\ & & \uparrow & & \uparrow \\ \mathrm{CE}(𝔞)& \leftarrow & \mathrm{CE}(𝔞{)}^I\\ \uparrow & & \uparrow & & \uparrow \\ ℝ& \leftarrow & W(𝔞)& \leftarrow & \mathrm{inv}(𝔞)\end{array}$$\array{ && CE(\mathfrak{a}) &\leftarrow& \mathbb{R} \\ && \uparrow && \uparrow \\ CE(\mathfrak{a}) &\leftarrow& CE(\mathfrak{a})^I \\ \uparrow &&\uparrow&& \uparrow \\ \mathbb{R} &\leftarrow& W(\mathfrak{a}) &\leftarrow& inv(\mathfrak{a}) }

The invariant polynomials in $\mathrm{inv}(𝔞)\subset \mathrm{CE}(\Sigma 𝔞)$ are those elements $P\in W(𝔞)$ in the Weil algebra that sit entirely in the shifted copy of $𝔞$ and whose differential sits entirely in the shifted copy.

See also invariant polynomial.

The $\mathrm{\infty }$-Chern-Weil sequence

The sequence

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

appearing here controls much of the $\mathrm{\infty }$-Chern-Weil theory, such as

• Cartan-Ehresmann ∞-connections

• symplectic ∞-Lie algebroids.

Transgression cocycles

We recall the procedure by which to an ∞-Lie algebroid invariant polynomial $\omega$ we associate an ∞-Lie algebroid cocycle $\nu$ that is in transgression with $\omega$.

The dg-algebra of invariant polynomial is a sub-dg-alghebra of the kernel of the morphism $W(𝔞)\to \mathrm{CE}(𝔞)$ from the Weil algebra to the Chevalley-Eilenberg algebra of $𝔞$

$$\mathrm{inv}(𝔞)\subset \mathrm{CE}(\Sigma 𝔞)=\ker (W(𝔞)\to \mathrm{CE}(𝔞)).$$inv(\mathfrak{a}) \subset CE(\Sigma \mathfrak{a}) = ker(W(\mathfrak{a}) \to CE(\mathfrak{a})) \,.

From the short exact sequence

$$\mathrm{CE}(\Sigma 𝔞)\to W(𝔞)\to \mathrm{CE}(𝔞)$$CE(\Sigma \mathfrak{a}) \to W(\mathfrak{a}) \to CE(\mathfrak{a})

we obtain the long exact sequence in cohomology

$$\cdots \to H^{n+1}(\mathrm{CE}(𝔞))\stackrel{\delta }{\to }{H}^{n+2}(\mathrm{CE}(\Sigma 𝔞))\to \cdots .$$\cdots \to H^{n+1}(CE(\mathfrak{a})) \stackrel{\delta}{\to} H^{n+2}(CE(\Sigma \mathfrak{a})) \to \cdots \,.

We say that $\mu \in \mathrm{CE}(𝔞)$ is in transgression with $\omega \in \mathrm{inv}(𝔞)\subset \mathrm{CE}(\Sigma 𝔞)$ if their classes map to each other under the connecting homomorphism $\delta$:

$$\delta :[\mu ]\mapsto [\omega ].$$\delta : [\mu] \mapsto [\omega] \,.

1. We first regard the invariant polynomial $\omega$ as an element of the Weil algebra $W(𝔞)$ under the inclusion $\mathrm{inv}(𝔞)\hookrightarrow W(𝔞)$, where, by the very definiton of invariant polynomials, it is closed: $d_{W(𝔞)}\omega =0$.

2. then we find an element ${\mathrm{cs}}_{\omega }\in W(𝔞)$ with the property that $d_{W(𝔞)}{\mathrm{cs}}_{\omega }=\omega$. This is guranteed to exist because $W(𝔞)$ has trivial cohomology.

3. then we send this element ${\mathrm{cs}}_{\omega }\in W(𝔞)$ along the restriction map $W(𝔞)\to \mathrm{CS}(𝔞)$ to an elemeent we call $\nu$.

The procedure is illustarted by the following diagram

$$\begin{array}{ccccc}0& & \omega & \leftarrow & \omega \\ {\uparrow }^{d_{\mathrm{CE}(𝔞)}}& & {\uparrow }^{d_{W(𝔞)}}\\ \nu & \leftarrow & \mathrm{cs}(\omega )\\ \\ \\ \\ \mathrm{CE}(𝔞)& \leftarrow & W(𝔞)& \leftarrow & \mathrm{inv}(𝔞)\end{array}$$\array{ 0 && \omega &\leftarrow & \omega \\ \;\;\uparrow^{\mathrlap{d_{CE(\mathfrak{a})}}} && \;\;\uparrow^{\mathrlap{d_{W(\mathfrak{a})}}} \\ \nu &\leftarrow& cs(\omega) \\ \\ \\ \\ CE(\mathfrak{a}) &\leftarrow& W(\mathfrak{a}) &\leftarrow& inv(\mathfrak{a}) }

From the fact that all morphisms involved respect the differential and from the fact that the image of $\omega$ in $\mathrm{CE}(𝔞)$ vanishes it follows that

• this element $\nu$ satisfies $d_{\mathrm{CE}(𝔞)}\nu =0$, hence that it is an $\mathrm{\infty }$-Lie algebroid cocycle.

• any two different choices of ${\mathrm{cs}}_{\omega }$ lead to cocylces $\mu$ that are cohomologous.

We say $\nu$ is a cocycle in transgression with $\omega$. We may call ${\mathrm{cs}}_{\omega }$ here a Chern-Simons element of $\omega$. Because for $A:TX\to 𝔞$ any collection of ∞-Lie algebroid valued differential forms coming dually from a dg-morphism ${\Omega }^{•}(X)\leftarrow W(𝔞):A$ the image $\omega (A)$ of $\omega$ will be a curvature characteristic form and the image ${\mathrm{cs}}_{\omega }(A)$ its corresponding Chern-Simons form.

In the case where $𝔤$ is an ordinary semisimple Lie algebra, this reduces to the ordinary study of ordinary Chern-Simons 3-forms associated with $𝔤$-valued 1-forms. This is described in the section Semisimple Lie algebras .