Contents

Contents

Idea

For every Lie algebra or ∞-Lie algebra or ∞-Lie algebroid $\mathfrak{a}$ there is its Chevalley-Eilenberg algebra $CE(\mathfrak{a})$ and its Weil algebra $W(\mathfrak{a})$ and a canonical dg-algebra morphism

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

Recall that a cocycle on $\mathfrak{a}$ is a closed element in $CE(\mathfrak{a})$. An invariant polynomial is a closed elements in $W(\mathfrak{a})$ that sits in the shifted copy $\wedge^\bullet (\mathfrak{a}^*)$.

This means that for $X \in \mathfrak{a}$, for $\iota_X : W(\mathfrak{a}) \to W(\mathfrak{a})$ the contraction derivation and $ad_X := [d_W, \iota_X]$ the corresponding Lie derivative, we have in particular that an invariant polynomial $\langle -\rangle \in W(\mathfrak{a})$ is invariant in the sense that

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

For $\mathfrak{a} = \mathfrak{g}$ an ordinary Lie algebra, an invariant polynomial on $\mathfrak{g}$ is precisely a symmetric multilinear map on $\mathfrak{g}$ which is $ad$-invariant in the ordinary sense.

Definition

Definition

For $\mathfrak{a}$ an ∞-Lie algebroid (of finite type, i.e. degreewise of finite rank) with Chevalley-Eilenberg algebra

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

and Weil algebra

$W(\mathfrak{a}) = (\wedge^\bullet (\mathfrak{a}^* \oplus \mathfrak{a}^*), d_{W(\mathfrak{a})})$

an invariant polynomial on $\mathfrak{a}$ is an elements $\langle - \rangle \in W(\mathfrak{a})$ with the property that

• $\langle - \rangle$ is a wedge product of generators in the shifted copy of $\mathfrak{a}^*$ $W(\mathfrak{a})$, i.e.

$\langle - \rangle \in \wedge^\bullet \mathfrak{a}^*$

or equivalently: for all $x \in \mathfrak{a}$ and $\iota_X : W(\mathfrak{a}) \to W(\mathfrak{a})$ the contraction derivation, we have

$\iota_x \langle -\rangle = 0 \,;$
• it is closed in $W(\mathfrak{a})$ in that $d_{W(\mathfrak{a})} \langle - \rangle = 0$

or more generally its differential is again in the shifted copy.

Remark

This implies that for

$ad_x := [d_{W(\mathfrak{a})}, \iota_X]$

the Lie derivative in $W(\mathfrak{a})$ along $x \in \mathfrak{a}$, which encodes the coadjoint action of $\mathfrak{a}$ on $W(\mathfrak{a})$, we have

$ad_x \langle - \rangle = 0$

for all $x$. But the condition for an invariant polynomial is stronger than these ad-invariances. For instance there are ∞-Lie algebra cocycles $\mu \in CE(\mathfrak{g})$ which when regarded as elements in $W(\mathfrak{g})$ are ad-invariant. But being entirely in the un-shifted copy, $\mu \in \wedge^\bullet \mathfrak{g}^*$, these are not invariant polynomials.

Definition

We say an invariant polynomial is decomposable if it is the wedge product in $W(\mathfrak{g})$ of two invariant polynomials of non-vanishing degree.

Definition

Two invariant polynomials $P_1, P_2 \in W(\mathfrak{g})$ are horizontally equivalent if there is $\omega \in ker(W(\mathfrak{g}) \to CE(\mathfrak{g}))$ such that

$P_1 = P_2 + d_W \omega \,.$
Proposition

Every decomposable invariant polynomial, def. , is horizontally equivalent to 0.

Proof

Let $P = P_1 \wedge P_2$ be a wedge product of two indecomposable polynomials. Then there exists a Chern-Simons element $cs_1 \in W(\mathfrak{g})$ such that $d_W cs_1 = P_1$. By the assumption that $P_2$ is in non-vanishing degree and hence in $ker(W(\mathfrak{g}) \to CE(\mathfrak{g}))$ it follows that

1. also $cs_1 \wedge P_2$ is in $ker(W(\mathfrak{g}) \to CE(\mathfrak{g}))$

2. $d_W (cs_1 \wedge P_2) = P_1 \wedge P_2$ .

Therefore $cs_1 \wedge P_1$ exhibits a horizontal equivalence $P_1 \wedge P_2 \sim 0$.

Observation

Horizontal equivalence classes of invariant polynomials on $\mathfrak{g}$ form a graded vector space $inv(\mathfrak{g})_V$. There is a morphism of graded vector spaces

$inv(\mathfrak{g})_V \hookrightarrow W(\mathfrak{g})$

unique up to horizontal equivalence, that sends each horizontal equivalence class to a representative.

Remark

By prop. it follows that $inv(\mathfrak{g})_V$ contains only indecomposable invariant polynomials.

Definition

We write $inv(\mathfrak{g})$ for the dg-algebra whose underlying graded algebra is the free graded algebra on the graded vector space $inv(\mathfrak{g})_V$, and whose differential is trivial.

Since invariant polynomials are closed, the inclusion of graded vector spaces from observation induces an inclusion (monomorphism) of dg-algebras

$inv(g) \hookrightarrow W(g) \,.$

Examples

On Lie algebras

Observation

For $\mathfrak{g}$ a Lie algebra, this definition of invariant polynomials is equivalent to more traditional ones.

Proof

To see this explicitly, let $\{t^a\}$ be a basis of $\mathfrak{g}^*$ and $\{r^a\}$ the corresponding basis of $\mathfrak{g}^*$. Write $\{C^a{}_{b c}\}$ for the structure constants of the Lie bracket in this basis.

Then for $P = P_{(a_1 , \cdots , a_k)} r^{a_1} \wedge \cdots \wedge r^{a_k} \in \wedge^{r} \mathfrak{g}^*$ an element in the shifted generators, the condition that it is $d_{W(\mathfrak{g})}$-closed is equivalent to

$C^{b}_{c (a_1} P_{b, \cdots, a_k)} t^c \wedge r^{a_1} \wedge \cdots \wedge r^{a_k} \,,$

where the parentheses around indices denotes symmetrization, as usual, so that this is equivalent to

$\sum_{i} C^{b}_{c (a_i} P_{a_1 \cdots a_{i-1} b a_{i+1} \cdots, a_k)} = 0$

for all choice of indices. This is the component-version of the familiar invariance statement

$\sum_i P(t_1, \cdots, t_{i-1}, [t_c, t_i], t_{i+1}, \cdots , t_k) = 0$

for all $t_\bullet \in \mathfrak{g}$.

On tangent Lie algebroids

For $X$ a smooth manifold, and invariant polynomial on the tangent Lie algebroid $\mathfrak{a} = T X$ is precisely a closed differential form on $X$.

On the string Lie 2-algebra

For $\mathfrak{g}$ a semisimple Lie algebra let $\mu_3 := \langle -,[-,-]\rangle$ be the canonical Lie algebra cocycle in degree 3, which is the one in transgression with the Killing form invariant polynomial $\langle -,-\rangle$.

Write $\mathfrak{g}_{\mu_3}$ for the corresponding string Lie 2-algebra. We have that the Chevalley-Eilenberg algebra $CE(\mathfrak{g}_{\mu_3})$ is given by

$d_{CE} t^a = - \frac{1}{2}C^a{}_{b c} t^b \wedge t^c$
$d_{CE} b = \mu_3$

and the Weil algebra $W(\mathfrak{g}_{\mu_3})$ is given by

$d_W t^a = - \frac{1}{2}C^a{}_{b c} t^b \wedge t^c + r^a$
$d_W b = \mu_3 - h$
$d_W r^a = - C^a_{b c} t^b \wedge r^c$
$d_W h = d_W \mu_3 = \sigma \mu_3 \,,$

where $\sigma$ acts by degree shift isomorphism on unshifted generators.

It follows at once that every invariant polynomial

$P = P_{a_1, \cdots, a_n} r^{a_1} \wedge \cdots \wedge r^{a_n}$

on the Lie algebra $\mathfrak{g}$ canonically identifies also with an invariant polynomial of the string Lie 2-algebra. But the differnce is that the Killing form $\langle -,- \rangle := P_{a b} r^a \wedge r^b$ is non-trivial as a polynomial on $\mathfrak{g}$, but as a polynomial on $\mathfrak{g}_{\mu_3}$ becomes horizontally equivalent ,def. ), to the trivial invariant polynomial.

Proposition

On the string Lie 2-algebra $\mathfrak{g}_{\mu_3}$ the Killing form $\langle -,-\rangle$ is horizontally equivalent to 0.

Proof

Let $cs_3 \in W(\mathfrak{g})$ be any Chern-Simons element for $\langle -,- \rangle$, hence an element such that

1. $cs_3|_{CE(\mathfrak{g})} = \mu_3$;

2. $d_W cs_3 = \langle -,- \rangle$.

Then notice that by the above we have in $W(\mathfrak{g}_{\mu_3})$ that the differential of the new generator $h$ is equal to that of $\mu_3$:

$d_W h = d_W \mu_3 \,.$

We on $\mathfrak{g}_{\mu_4}$ we can replace $\mu_3$ by $h$ and still get a Chern-Simons element for the Killing form:

$\tilde cs_3 := cs_3 - \mu_3 + h \,.$
$d_W \tilde cs_3 = \langle -,- \rangle \,.$

But while $\mu_3$ is not in $ker(W(\mathfrak{g}_{\mu_3}) \to CE(\mathfrak{g}_{\mu_3}))$, the element $h$ is, by definition. Therefore $\tilde cs_3$ is in that kernel, and hence exhibits a horizontal equivalence between $\langle -,- \rangle$ and $0$.

This is a special case of the more general statement below, about invariant polynomials on shifted central extensions.

For illustration purposes it is useful to consider the following variant of this example:

Definition

Write

$(b \mathbb{R} \to \mathfrak{string}) \in L_\infty Alg$

for the L-∞ algebra defined by the fact that its Chevalley-Eilenberg algebra is given by

$d_{CE} t^a = - \frac{1}{2} C^a{}_{b c} t^b \wedge t^c$
$d_{CE} b = \mu_3 - c$
$d_{CE} c = 0 \,,$

where $\{t^a\}$ is a dual basis in degree 1 for some semisimple Lie algebra $\mathfrak{g}$ as above, $b$ and $c$ are generators in degree 2 and 3, respectively, and $\mu_3 \propto \langle -,[-,-]\rangle$ is the canonical Lie algebra cocycle in degree 3, as above.

It is easily seen that

Observation

The canonical morphism

$\mathfrak{g} \to (b \mathbb{R} \to \mathfrak{string})$

given dually by sending

$t^a \mapsto t^a\,,\,\,\, b \mapsto 0\,,\, \,\, c \mapsto \mu_3$

is a weak equivalence.

So the Lie 3-algebra $(b \mathbb{R} \to \mathfrak{string})$ is a kind of resolution of the ordinary Lie algebra $\mathfrak{g}$. It is for instance of use in the presentation of twisted differential string structures, where the shifted piece $b \mathbb{R}$ in $(b \mathbb{R} \to \mathfrak{string})$ picks up the failure of $\mathfrak{so}$-valued connections to lift to $\mathfrak{string}$-2-connections.

The proof of the following proposition may be instructive for seeing how the definition of horizontal equivalence of invariant polynomials takes care of having the invariant polynomials of $(b\mathbb{R} \to \mathfrak{string})$ agree with those of $\mathfrak{g}$.

Observation

There is an isomorphism

$inv(\mathfrak{g}) \simeq inv(b \mathbb{R} \to \mathfrak{string}) \,.$
Proof

Notice that the Weil algebra of $(b\mathbb{R} \to \mathfrak{string})$ is given by

$d_W t^a = - \frac{1}{2} C^a{}_{b c} t^b \wedge t^c + r^a$
$d_W b = \mu_3 - c - h$
$d_W c = g$

for new generators $\{r^a\}$ in degree 2, $h$ in degree 3 and $g$ in degree 4, coming with their Bianchi identities

$d_W r^a = - C^a{}_{b c}t^b \wedge r^c$
$d_W h = d_W \mu_3 - g$
$d_W g = 0$

For the following computations let $\{k_{a b}\}$ be the structure constants of the Killing form, so that

$\langle -,- \rangle = k_{a b} r^a \wedge r^b$

and assume that $\mu_3$ is normalized such that

$\mu_3 = k_{a a'}C^{a'}_{b c} t^a \wedge t^b \wedge t^c$

(if another normalization is chosen, then the corresponding factor will float around the following formulas without changing anything of the end result).

Now the indecomposable invariant polynomials are those of $\mathfrak{g}$ and one additional one: $g$. This means that before deviding out horizontal equivalence on generators, the invariant polynomials of $(b \mathbb{R} \to \mathfrak{string})$ are not equal to those of $\mathfrak{g}$, due to the superfluous generator $g$.

But we do have the horizontal equivalence relation

$\langle -,-\rangle = g + d_W (cs - \mu_3 + h) \,,$

where $cs$ is any Chern-Simons element for $\langle - , \rangle$, for instance

$cs = \frac{1}{6} k_{a a'}C^{a'}_{b c} t^a \wedge t^b \wedge t^c + k_{a b} t^a r^b \,,$

Notice that the homotopy $cs - \mu_3 + h$ here is indeed in $ker(W(\mathfrak{g}) \to CE(\mathfrak{g}))$: the component of $cs$ not in that kernel is precisely $\mu$. The above formula subtracts this offending summand and replaces it with the new generator $h$, which by definition is in the kernel and whose image under $d_W$ is the image of $\mu$ under $d_W$, plus the superfluous new generator of invariant polynomials.

Therefore in horizontal equivalence classes of invariant polynomials on $(b \mathbb{R} \to \mathfrak{string})$ the superfluous $g$ is identified with the Killing form $\langle-,- \rangle$, and hence the claim follows.

On symplectic Lie $n$-algebroids

A symplectic Lie n-algebroid is an L-infinity algebroid that carries a binary and non-degeneraty invariant polynomial of grade $n$. This is a generalization of the notion of symplectic form to which it reduces for $n = 0$.

Properties

As differential forms on the moduli stack of connections

The invariant polynomials of a Lie algebra $\mathfrak{g}$, thought of as equipped with trivial differential, are the de Rham complex of differential forms on the universal moduli stack $\mathbf{B}G_{conn}$ of $G$-principal connections Freed-Hopkins 13.

$\mathrm{inv}(\mathfrak{g})\simeq \Omega^\bullet(\mathbf{B}G_{conn}) \,.$

For more on this see also at Weil algebra – Characterization in the smooth infinity-topos.

On reductive Lie algebras

Proposition

Let $\mathfrak{g}$ be a reductive Lie algebra. Then the subalgebra of invariant polynomials in the Weil algebra is the free graded algebra on the graded vector space of indecomposable invariant polynomials.

This graded vector space has a vector space isomorphism of degree -1 to the graded vector space of odd generators of the Lie algebra cohomology $H^\bullet(\mathfrak{g}) = H^\bullet(CE(\mathfrak{g}))$.

This appears for instance as (GHV, vol III, page 242, theorem I).

Role in $\infty$-Chern-Weil theory

In ($\infty$-)Chern-Weil theory the crucial role played by the invariant polynomials is their relation to ∞-Lie algebra cocycles. One may understand invariant invariant polynomials as extending under Lie integration $\infty$-Lie algebra cocycles from cohomology to differential cohomology.

Transgression cocycles and Chern-Simons elements

Definition

(Chern-Simons elements and transgression cocycles)

Let $\mathfrak{a} = \mathfrak{g}$ be an ∞-Lie algebra. Since the cochain cohomology of the Weil algebra $W(\mathfrak{g})$ is trivial, for every invariant polynomial $\langle -\rangle \in W(\mathfrak{g})$ there is necessarily an element $cs \in W(\mathfrak{g})$ with

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

This we call a Chern-Simons element for $\langle -\rangle$.

This element $cs$ will in general not sit entirely in the shifted copy. Its restriction

$\mu := cs|_{\wedge^\bullet \mathfrak{g}^*} \in CE(\mathfrak{g})$

is a ∞-Lie algebra cocycle. We say this is in transgression with $\langle -\rangle$.

In total this construction yields a commuting diagram

$\array{ CE(\mathfrak{g}) &\stackrel{\mu}{\leftarrow}& CE(b^{n-1}\mathbb{R}) &&& cocycle \\ \uparrow && \uparrow \\ W(\mathfrak{g}) &\stackrel{(cs,\langle -\rangle)}{\leftarrow}& W(b^{n-1} \mathbb{R}) &&& Chern-Simons-element \\ \uparrow && \uparrow \\ inv(\mathfrak{g}) &\stackrel{\langle -\rangle}{\leftarrow}& CE(b^n \mathbb{R}) &&& invariant\; polynomial } \,,$

where $b^{n-1}\mathbb{R}$ denotes the ∞-Lie algebra whose CE-algebra has a single generator in degree $n$ and vanishing differential, and where $CE(b^n \mathbb{R}) = inv(b^{n-1}\mathbb{R})$ is the algebra of invariant polynomials of $b^{n-1} \mathbb{R}$.

Proposition

The element $\mu \in CE(\mathfrak{g})$ associated to an invariant polynomial $\langle -\rangle$ by the above procedure is indeed a cocycle, and its cohomology class is independent of the choice of the element $cs$ involved.

Proof

The procedure that assigns $\mu$ to $\langle- \rangle$ is illustarted by the following diagram

$\array{ 0 && \langle-\rangle &\leftarrow & \langle-\rangle \\ \;\;\uparrow^{\mathrlap{d_{CE(\mathfrak{g})}}} && \;\;\uparrow^{\mathrlap{d_{W(\mathfrak{g})}}} \\ \mu &\leftarrow& cs \\ \\ \\ \\ CE(\mathfrak{a}) &\leftarrow& W(\mathfrak{a}) &\leftarrow& inv(\mathfrak{a}) }$

From the fact that all morphisms involved respect the differential and from the fact that the image of $\langle-\rangle$ in $CE(\mathfrak{g})$ vanishes it follows that

• the element $\mu$ satisfies $d_{CE(\mathfrak{a})} \mu = 0$, hence that it is an ∞-Lie algebra cocycle;

• any two different choices of $cs$ lead to cocylces $\mu$ that are cohomologous.

This construction exhibits effectively the preimage of the connecting homomorphism in the cochain cohomology sequence induced by $W(\mathfrak{g}) \to CE(\mathfrak{g})$:

The dg-algebra of invariant polynomials is a sub-dg-algebra of the kernel of the morphism $i^* : W(\mathfrak{a}) \to CE(\mathfrak{a})$ from the Weil algebra to the Chevalley-Eilenberg algebra of $\mathfrak{a}$

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

From the short exact sequence

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

we obtain the long exact sequence in cohomology

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

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

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

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

Chern-Simons and curvature characteristic forms

For $\mathfrak{g}$ a Lie n-alghebra, let $\mathbf{B}G := \mathbf{cosk}_{n+1} \exp(\mathfrak{g})$ be the ∞-Lie group obtained by Lie integration from it.

For $X$ a paracompact smooth manifold with good open cover $\{U_i \to X\}$ whose Cech nerve we write $C(U)$, a cocycle for a $G$-principal ∞-bundle on $X$ is cocycle with coefficients in the simplicial sheaf

$\mathbf{B}G = \mathbf{cosk}_{n+1}((U,[k]) \mapsto \{ \Omega^\bullet_{vert}(U \times \Delta^k) \leftarrow CE(\mathfrak{g}) \}) \,.$

We say an $\infty$-connection on this is an extension to a cocycle with coefficients in the simplicial sheaf

$\mathbf{B}G_{diff} = \mathbf{cosk}_{n+1}((U,[k]) \mapsto \left\{ \array{ \Omega^\bullet_{vert}(U \times \Delta^k) &\leftarrow& CE(\mathfrak{g}) &&& underlying \; cocycle \\ \uparrow && \uparrow \\ \Omega^\bullet(U\times \Delta^k) &\stackrel{}{\leftarrow}& W(\mathfrak{g}) &&& connection } \right\} \,.$

The diagrams on the left encode those $\mathfrak{g}$-valued forms on $U \times \Delta^k$ whose curvature vanishes on $\Delta^k$. One can show that one can always find a genuine $\infty$-connection: one for which the curvatures have no leg along $\Delta^k$, in that they land in $\Omega^\bullet(U) \otimes C^\infty(\Delta^k)$. For those the above diagram extends to

$\array{ \Omega^\bullet(U \times \Delta^k)_{vert} &\leftarrow& CE(\mathfrak{g}) &&& underlying \; cocycle \\ \uparrow && \uparrow \\ \Omega^\bullet(U\times \Delta^k) &\stackrel{}{\leftarrow}& W(\mathfrak{g}) &&& connection \\ \uparrow && \uparrow \\ \Omega^\bullet(U) &\leftarrow& inv(\mathfrak{g}) &&& curvature } \,.$

This defines the simplicial presheaf that classifies connections on ∞-bundles.

By pasting-postcomposition with the above diagrams for an invariant polynomial we obtain connections with values in $b^n \mathbb{R}$

$\array{ \Omega^\bullet_{vert}(U \times \Delta^k) &\leftarrow& CE(\mathfrak{g}) &\stackrel{\mu}{\leftarrow}& CE(b^{n-1}\mathbb{R}) &&& underlying \; cocycle \\ \uparrow && \uparrow && \uparrow \\ \Omega^\bullet(U\times \Delta^k) &\stackrel{}{\leftarrow}& W(\mathfrak{g}) &\stackrel{(cs,\langle- \rangle)}{\leftarrow}& W(b^{n-1}\mathbb{R}) &&& Chern-Simons forms \\ \uparrow && \uparrow && \uparrow \\ \Omega^\bullet(U) &\leftarrow& inv(\mathfrak{g}) &\stackrel{\langle -\rangle}{\leftarrow}& CE(b^n \mathbb{R}) &&& curvature\;characteristic\;form } \,,$

where in the bottom row we have the curvature characteristic forms $\langle F_\nabla\rangle$ coresponding to the connection, and in the middle the corresponding Chern-Simons forms.

More details for the moment at ∞-Chern-Weil theory introduction.

Invariant polynomials for Lie algebras of simple Lie groups are disussed in

A standard textbook account of the traditional theory is in volume III of

The notion of invariant polynomials of $L_\infty$-algebras has been introduced in

The abstract characterization is due to

An account in the more general context of Lie theory in cohesive (infinity,1)-toposes is in section 3.3.11 of