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

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

Recall that a cocycle on 𝔞\mathfrak{a} is a closed element in CE(𝔞)CE(\mathfrak{a}). An invariant polynomial is a closed elements in W(𝔞)W(\mathfrak{a}) that sits in the shifted copy (𝔞 *[1])\wedge^\bullet (\mathfrak{a}^*[1]).

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

ad X=0. ad_X \langle -\rangle = 0 \,.

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



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

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

and Weil algebra

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

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

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

    𝔞 *[1] \langle - \rangle \in \wedge^\bullet \mathfrak{a}^*[1]

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

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

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


This implies that for

ad x:=[d W(𝔞),ι X] ad_x := [d_{W(\mathfrak{a})}, \iota_X]

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

ad x=0 ad_x \langle - \rangle = 0

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


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


Two invariant polynomials P 1,P 2W(𝔤)P_1, P_2 \in W(\mathfrak{g}) are horizontally equivalent if there is ωker(W(𝔤)CE(𝔤))\omega \in ker(W(\mathfrak{g}) \to CE(\mathfrak{g})) such that

P 1=P 2+d Wω. P_1 = P_2 + d_W \omega \,.

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


Let P=P 1P 2P = P_1 \wedge P_2 be a wedge product of two indecomposable polynomials. Then there exists a Chern-Simons element cs 1W(𝔤)cs_1 \in W(\mathfrak{g}) such that d Wcs 1=P 1d_W cs_1 = P_1. By the assumption that P 2P_2 is in non-vanishing degree and hence in ker(W(𝔤)CE(𝔤))ker(W(\mathfrak{g}) \to CE(\mathfrak{g})) it follows that

  1. also cs 1P 2cs_1 \wedge P_2 is in ker(W(𝔤)CE(𝔤))ker(W(\mathfrak{g}) \to CE(\mathfrak{g}))

  2. d W(cs 1P 2)=P 1P 2d_W (cs_1 \wedge P_2) = P_1 \wedge P_2 .

Therefore cs 1P 1cs_1 \wedge P_1 exhibits a horizontal equivalence P 1P 20P_1 \wedge P_2 \sim 0.


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

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

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


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


We write inv(𝔤)inv(\mathfrak{g}) for the dg-algebra whose underlying graded algebra is the free graded algebra on the graded vector space inv(𝔤) Vinv(\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)W(g). inv(g) \hookrightarrow W(g) \,.


On Lie algebras


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


To see this explicitly, let {t a}\{t^a\} be a basis of 𝔤 *\mathfrak{g}^* and {r a}\{r^a\} the corresponding basis of 𝔤 *[1]\mathfrak{g}^*[1]. Write {C a bc}\{C^a{}_{b c}\} for the structure constants of the Lie bracket in this basis.

Then for P=P (a 1,,a k)r a 1r a k r𝔤 *[1]P = P_{(a_1 , \cdots , a_k)} r^{a_1} \wedge \cdots \wedge r^{a_k} \in \wedge^{r} \mathfrak{g}^*[1] an element in the shifted generators, the condition that it is d W(𝔤)d_{W(\mathfrak{g})}-closed is equivalent to

C c(a 1 bP b,,a k)t cr a 1r a k, 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

iC c(a i bP a 1a i1ba i+1,a k)=0 \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

iP(t 1,,t i1,[t c,t i],t i+1,,t k)=0 \sum_i P(t_1, \cdots, t_{i-1}, [t_c, t_i], t_{i+1}, \cdots , t_k) = 0

for all t 𝔤t_\bullet \in \mathfrak{g}.

On semisimple Lie algebras

See Killing form, string Lie 2-algebra.

On tangent Lie algebroids

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

On the string Lie 2-algebra

For 𝔤\mathfrak{g} a semisimple Lie algebra let μ 3:=,[,]\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 𝔤 μ 3\mathfrak{g}_{\mu_3} for the corresponding string Lie 2-algebra. We have that the Chevalley-Eilenberg algebra CE(𝔤 μ 3)CE(\mathfrak{g}_{\mu_3}) is given by

d CEt a=12C a bct bt c d_{CE} t^a = - \frac{1}{2}C^a{}_{b c} t^b \wedge t^c
d CEb=μ 3 d_{CE} b = \mu_3

and the Weil algebra W(𝔤 μ 3)W(\mathfrak{g}_{\mu_3}) is given by

d Wt a=12C a bct bt c+r a d_W t^a = - \frac{1}{2}C^a{}_{b c} t^b \wedge t^c + r^a
d Wb=μ 3h d_W b = \mu_3 - h
d Wr a=C bc at br c d_W r^a = - C^a_{b c} t^b \wedge r^c
d Wh=d Wμ 3=σμ 3, d_W h = d_W \mu_3 = \sigma \mu_3 \,,

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

It follows at once that every invariant polynomial

P=P a 1,,a nr a 1r a n P = P_{a_1, \cdots, a_n} r^{a_1} \wedge \cdots \wedge r^{a_n}

on the Lie algebra 𝔤\mathfrak{g} canonically identifies also with an invariant polynomial of the string Lie 2-algebra. But the differnce is that the Killing form ,:=P abr ar b\langle -,- \rangle := P_{a b} r^a \wedge r^b is non-trivial as a polynomial on 𝔤\mathfrak{g}, but as a polynomial on 𝔤 μ 3\mathfrak{g}_{\mu_3} becomes horizontally equivalent ,def. ), to the trivial invariant polynomial.


On the string Lie 2-algebra 𝔤 μ 3\mathfrak{g}_{\mu_3} the Killing form ,\langle -,-\rangle is horizontally equivalent to 0.


Let cs 3W(𝔤)cs_3 \in W(\mathfrak{g}) be any Chern-Simons element for ,\langle -,- \rangle, hence an element such that

  1. cs 3| CE(𝔤)=μ 3cs_3|_{CE(\mathfrak{g})} = \mu_3;

  2. d Wcs 3=,d_W cs_3 = \langle -,- \rangle.

Then notice that by the above we have in W(𝔤 μ 3)W(\mathfrak{g}_{\mu_3}) that the differential of the new generator hh is equal to that of μ 3\mu_3:

d Wh=d Wμ 3. d_W h = d_W \mu_3 \,.

We on 𝔤 μ 4\mathfrak{g}_{\mu_4} we can replace μ 3\mu_3 by hh and still get a Chern-Simons element for the Killing form:

cs˜ 3:=cs 3μ 3+h. \tilde cs_3 := cs_3 - \mu_3 + h \,.
d Wcs˜ 3=,. d_W \tilde cs_3 = \langle -,- \rangle \,.

But while μ 3\mu_3 is not in ker(W(𝔤 μ 3)CE(𝔤 μ 3))ker(W(\mathfrak{g}_{\mu_3}) \to CE(\mathfrak{g}_{\mu_3})), the element hh is, by definition. Therefore cs˜ 3\tilde cs_3 is in that kernel, and hence exhibits a horizontal equivalence between ,\langle -,- \rangle and 00.

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:



(b𝔰𝔱𝔯𝔦𝔫𝔤)L Alg (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 CEt a=12C a bct bt c d_{CE} t^a = - \frac{1}{2} C^a{}_{b c} t^b \wedge t^c
d CEb=μ 3c d_{CE} b = \mu_3 - c
d CEc=0, d_{CE} c = 0 \,,

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

It is easily seen that


The canonical morphism

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

given dually by sending

t at a,b0,cμ 3 t^a \mapsto t^a\,,\,\,\, b \mapsto 0\,,\, \,\, c \mapsto \mu_3

is a weak equivalence.

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


There is an isomorphism

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

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

d Wt a=12C a bct bt c+r a d_W t^a = - \frac{1}{2} C^a{}_{b c} t^b \wedge t^c + r^a
d Wb=μ 3ch d_W b = \mu_3 - c - h
d Wc=g d_W c = g

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

d Wr a=C a bct br c d_W r^a = - C^a{}_{b c}t^b \wedge r^c
d Wh=d Wμ 3g d_W h = d_W \mu_3 - g
d Wg=0 d_W g = 0

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

,=k abr ar b \langle -,- \rangle = k_{a b} r^a \wedge r^b

and assume that μ 3\mu_3 is normalized such that

μ 3=k aaC bc at at bt c \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 𝔤\mathfrak{g} and one additional one: gg. This means that before deviding out horizontal equivalence on generators, the invariant polynomials of (b𝔰𝔱𝔯𝔦𝔫𝔤)(b \mathbb{R} \to \mathfrak{string}) are not equal to those of 𝔤\mathfrak{g}, due to the superfluous generator gg.

But we do have the horizontal equivalence relation

,=g+d W(csμ 3+h), \langle -,-\rangle = g + d_W (cs - \mu_3 + h) \,,

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

cs=16k aaC bc at at bt c+k abt ar b, 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 csμ 3+hcs - \mu_3 + h here is indeed in ker(W(𝔤)CE(𝔤))ker(W(\mathfrak{g}) \to CE(\mathfrak{g})): the component of cscs not in that kernel is precisely μ\mu. The above formula subtracts this offending summand and replaces it with the new generator hh, which by definition is in the kernel and whose image under d Wd_W is the image of μ\mu under d Wd_W, plus the superfluous new generator of invariant polynomials.

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

On symplectic Lie nn-algebroids

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


As differential forms on the moduli stack of connections

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

inv(𝔤)Ω (BG conn). \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


Let 𝔤\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 (𝔤)=H (CE(𝔤))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


(Chern-Simons elements and transgression cocycles)

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

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

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

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

μ:=cs| 𝔤 *CE(𝔤) \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

CE(𝔤) μ CE(b n1) cocycle W(𝔤) (cs,) W(b n1) ChernSimonselement inv(𝔤) CE(b n) invariantpolynomial, \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 n1b^{n-1}\mathbb{R} denotes the ∞-Lie algebra whose CE-algebra has a single generator in degree nn and vanishing differential, and where CE(b n)=inv(b n1)CE(b^n \mathbb{R}) = inv(b^{n-1}\mathbb{R}) is the algebra of invariant polynomials of b n1b^{n-1} \mathbb{R}.


The element μCE(𝔤)\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 cscs involved.


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

0 d CE(𝔤) d W(𝔤) μ cs CE(𝔞) W(𝔞) inv(𝔞) \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(𝔤)CE(\mathfrak{g}) vanishes it follows that

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

  • any two different choices of cscs lead to cocylces μ\mu that are cohomologous.

This construction exhibits effectively the preimage of the connecting homomorphism in the cochain cohomology sequence induced by W(𝔤)CE(𝔤)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(𝔞)CE(𝔞)i^* : W(\mathfrak{a}) \to CE(\mathfrak{a}) from the Weil algebra to the Chevalley-Eilenberg algebra of 𝔞\mathfrak{a}

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

From the short exact sequence

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

we obtain the long exact sequence in cohomology

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

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

δ:[μ][ω]. \delta : [\mu] \mapsto [\omega] \,.

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

Chern-Simons and curvature characteristic forms

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

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

BG=cosk n+1((U,[k]){Ω vert (U×Δ k)CE(𝔤)}). \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

BG diff=cosk n+1((U,[k]){Ω vert (U×Δ k) CE(𝔤) underlyingcocycle Ω (U×Δ k) W(𝔤) connection}. \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 𝔤\mathfrak{g}-valued forms on U×Δ kU \times \Delta^k whose curvature vanishes on Δ k\Delta^k. One can show that one can always find a genuine \infty-connection: one for which the curvatures have no leg along Δ k\Delta^k, in that they land in Ω (U)C (Δ k)\Omega^\bullet(U) \otimes C^\infty(\Delta^k). For those the above diagram extends to

Ω (U×Δ k) vert CE(𝔤) underlyingcocycle Ω (U×Δ k) W(𝔤) connection Ω (U) inv(𝔤) curvature. \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 nb^n \mathbb{R}

Ω vert (U×Δ k) CE(𝔤) μ CE(b n1) underlyingcocycle Ω (U×Δ k) W(𝔤) (cs,) W(b n1) ChernSimonsforms Ω (U) inv(𝔤) CE(b n) curvaturecharacteristicform, \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 F \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.


The idea of invariant polynomials as a GG-invariant subalgebra in the Weil algebra, and their use in what is now called Chern-Weil theory, originates with

  • André Weil, Géométrie différentielle des espaces fibres, unpublished, item [1949e] in: André Weil Oeuvres Scientifiques / Collected Papers, vol. 1 (1926-1951), 422-436, Springer 2009 (ISBN:978-3-662-45256-1)

  • Henri Cartan, Cohomologie réelle d’un espace fibré principal différentiable. I : notions d’algèbre différentielle, algèbre de Weil d’un groupe de Lie, Séminaire Henri Cartan, Volume 2 (1949-1950), Talk no. 19, May 1950 (numdam:SHC_1949-1950__2__A18_0)

    Henri Cartan, Notions d’algèbre différentielle; applications aux groupes de Lie et aux variétés où opère un groupe de Lie, in: Centre Belge de Recherches Mathématiques, Colloque de Topologie (Espaces Fibrés) Tenu à Bruxelles du 5 au 8 juin 1950, Georges Thon 1951 (GoogleBooks)

    (These two articles have the same content, with the same section outline, but not the same wording. The first one is a tad more detailed.)

  • Shiing-shen Chern, Differential geometry of fiber bundles, in: Proceedings of the International Congress of Mathematicians, Cambridge, Mass., (August-September 1950), vol. 2, pages 397-411, Amer. Math. Soc., Providence, R. I. (1952) (pdf, full proceedings vol 2 pdf)

  • Shoshichi Kobayashi, Katsumi Nomizu, Section XII.2 in: Foundations of Differential Geometry, Volume 1, Wiley 1963 (web, ISBN:9780471157335, Wikipedia)

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 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

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