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\begin{document}

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\section*{invariant polynomial}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{lie_theory}{}\paragraph*{{$\infty$-Lie theory}}\label{lie_theory}

[[!include infinity-Lie theory - contents]]

\hypertarget{chernweil_theory}{}\paragraph*{{$\infty$-Chern-Weil theory}}\label{chernweil_theory}

[[!include infinity-Chern-Weil theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{OnLieAlgebras}{On Lie algebras}\dotfill \pageref*{OnLieAlgebras} \linebreak
\noindent\hyperlink{SemisimpLie}{On semisimple Lie algebras}\dotfill \pageref*{SemisimpLie} \linebreak
\noindent\hyperlink{on_tangent_lie_algebroids}{On tangent Lie algebroids}\dotfill \pageref*{on_tangent_lie_algebroids} \linebreak
\noindent\hyperlink{OnStringLie2Algebra}{On the string Lie 2-algebra}\dotfill \pageref*{OnStringLie2Algebra} \linebreak
\noindent\hyperlink{on_symplectic_lie_algebroids}{On symplectic Lie $n$-algebroids}\dotfill \pageref*{on_symplectic_lie_algebroids} \linebreak
\noindent\hyperlink{Properties}{Properties}\dotfill \pageref*{Properties} \linebreak
\noindent\hyperlink{AsFormsOnBGconn}{As differential forms on the moduli stack of connections}\dotfill \pageref*{AsFormsOnBGconn} \linebreak
\noindent\hyperlink{OnReductiveLieAlg}{On reductive Lie algebras}\dotfill \pageref*{OnReductiveLieAlg} \linebreak
\noindent\hyperlink{role_in_chernweil_theory}{Role in $\infty$-Chern-Weil theory}\dotfill \pageref*{role_in_chernweil_theory} \linebreak
\noindent\hyperlink{TransgressionCocycles}{Transgression cocycles and Chern-Simons elements}\dotfill \pageref*{TransgressionCocycles} \linebreak
\noindent\hyperlink{chernsimons_and_curvature_characteristic_forms}{Chern-Simons and curvature characteristic forms}\dotfill \pageref*{chernsimons_and_curvature_characteristic_forms} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{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

\begin{displaymath}
CE(\mathfrak{a}) \leftarrow W(\mathfrak{a})
  \,.
\end{displaymath}
Recall that a [[∞-Lie algebra cohomology|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 $\wedge^\bullet (\mathfrak{a}^*[1])$.

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

\begin{displaymath}
ad_X \langle -\rangle = 0
  \,.
\end{displaymath}
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.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

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

\begin{displaymath}
CE(\mathfrak{g}) = (\wedge^\bullet \mathfrak{a}^*, d_{CE(\mathfrak{a}}))
\end{displaymath}
and [[Weil algebra]]

\begin{displaymath}
W(\mathfrak{a}) = (\wedge^\bullet (\mathfrak{a}^* \oplus \mathfrak{a}^*[1]), d_{W(\mathfrak{a})})
\end{displaymath}
an \textbf{invariant polynomial} on $\mathfrak{a}$ is an elements $\langle - \rangle \in W(\mathfrak{a})$ with the property that

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

\begin{displaymath}
\langle - \rangle \in \wedge^\bullet \mathfrak{a}^*[1]
\end{displaymath}
or equivalently: for all $x \in \mathfrak{a}$ and $\iota_X : W(\mathfrak{a}) \to W(\mathfrak{a})$ the contraction [[derivation]], we have

\begin{displaymath}
\iota_x \langle -\rangle = 0
  \,;
\end{displaymath}

\item 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.



\end{itemize}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
This implies that for

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

\begin{displaymath}
ad_x \langle - \rangle = 0
\end{displaymath}
for all $x$. But the condition for an invariant polynomial is stronger than these ad-invariances. For instance there are [[∞-Lie algebra cohomology|∞-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.

\end{remark}
\begin{defn}
\label{DecomposableInvPolynomial}\hypertarget{DecomposableInvPolynomial}{}
We say an invariant polynomial is \emph{decomposable} if it is the wedge product in $W(\mathfrak{g})$ of two invariant polynomials of non-vanishing degree.

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

\begin{displaymath}
P_1 = P_2 + d_W \omega
  \,.
\end{displaymath}
\end{defn}
\begin{prop}
\label{DecomposablePolysAreHorizontallyTrivial}\hypertarget{DecomposablePolysAreHorizontallyTrivial}{}
Every decomposable invariant polynomial, def. \ref{DecomposableInvPolynomial}, is horizontally equivalent to 0.

\end{prop}
\begin{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

\begin{enumerate}%
\item also $cs_1 \wedge P_2$ is in $ker(W(\mathfrak{g}) \to CE(\mathfrak{g}))$


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



\end{enumerate}
Therefore $cs_1 \wedge P_1$ exhibits a horizontal equivalence $P_1 \wedge P_2 \sim 0$.

\end{proof}
\begin{prop}
\label{ClassesOfInvPolysFormGradedVectorSpace}\hypertarget{ClassesOfInvPolysFormGradedVectorSpace}{}
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

\begin{displaymath}
inv(\mathfrak{g})_V \hookrightarrow W(\mathfrak{g})
\end{displaymath}
unique up to horizontal equivalence, that sends each horizontal equivalence class to a representative.

\end{prop}
\begin{remark}
\label{}\hypertarget{}{}
By prop. \ref{DecomposablePolysAreHorizontallyTrivial} it follows that $inv(\mathfrak{g})_V$ contains only indecomposable invariant polynomials.

\end{remark}
\begin{defn}
\label{}\hypertarget{}{}
We write $inv(\mathfrak{g})$ for the [[dg-algebra]] whose underlying [[graded algebra]] is the [[free construction|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 \ref{ClassesOfInvPolysFormGradedVectorSpace} induces an inclusion ([[monomorphism]]) of dg-algebras

\begin{displaymath}
inv(g) \hookrightarrow W(g)
  \,.
\end{displaymath}
\end{defn}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{OnLieAlgebras}{}\subsubsection*{{On Lie algebras}}\label{OnLieAlgebras}

\begin{lemma}
\label{}\hypertarget{}{}
For $\mathfrak{g}$ a [[Lie algebra]], this definition of invariant polynomials is equivalent to more traditional ones.

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

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

\begin{displaymath}
C^{b}_{c (a_1} P_{b, \cdots, a_k)} t^c \wedge r^{a_1} \wedge \cdots \wedge r^{a_k}
  \,,
\end{displaymath}
where the parentheses around indices denotes symmetrization, as usual, so that this is equivalent to

\begin{displaymath}
\sum_{i} C^{b}_{c (a_i} P_{a_1 \cdots a_{i-1} b a_{i+1} \cdots, a_k)}
  = 0
\end{displaymath}
for all choice of indices. This is the component-version of the familiar invariance statement

\begin{displaymath}
\sum_i P(t_1, \cdots, t_{i-1}, [t_c, t_i], t_{i+1}, \cdots , t_k)
  = 0
\end{displaymath}
for all $t_\bullet \in  \mathfrak{g}$.

\end{proof}
\hypertarget{SemisimpLie}{}\subsubsection*{{On semisimple Lie algebras}}\label{SemisimpLie}

See [[Killing form]], [[string Lie 2-algebra]].

\hypertarget{on_tangent_lie_algebroids}{}\subsubsection*{{On tangent Lie algebroids}}\label{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 $X$.

\hypertarget{OnStringLie2Algebra}{}\subsubsection*{{On the string Lie 2-algebra}}\label{OnStringLie2Algebra}

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

\begin{displaymath}
d_{CE} t^a = - \frac{1}{2}C^a{}_{b c} t^b \wedge t^c
\end{displaymath}
\begin{displaymath}
d_{CE} b = \mu_3
\end{displaymath}
and the [[Weil algebra]] $W(\mathfrak{g}_{\mu_3})$ is given by

\begin{displaymath}
d_W t^a = - \frac{1}{2}C^a{}_{b c} t^b \wedge t^c + r^a
\end{displaymath}
\begin{displaymath}
d_W b = \mu_3 - h
\end{displaymath}
\begin{displaymath}
d_W r^a = - C^a_{b c} t^b \wedge r^c
\end{displaymath}
\begin{displaymath}
d_W h = d_W \mu_3 = \sigma \mu_3
  \,,
\end{displaymath}
where $\sigma$ acts by degree shift isomorphism on unshifted generators.

It follows at once that every invariant polynomial

\begin{displaymath}
P = P_{a_1, \cdots, a_n} r^{a_1} \wedge \cdots \wedge r^{a_n}
\end{displaymath}
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. \ref{HorizontalEquivalence}), to the trivial invariant polynomial.

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

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

\begin{enumerate}%
\item $cs_3|_{CE(\mathfrak{g})} = \mu_3$;


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



\end{enumerate}
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 $\mu_3$:

\begin{displaymath}
d_W h = d_W \mu_3
  \,.
\end{displaymath}
We on $\mathfrak{g}_{\mu_4}$ we can replace $\mu_3$ by $h$ and still get a [[Chern-Simons element]] for the Killing form:

\begin{displaymath}
\tilde cs_3 := cs_3 - \mu_3 + h
  \,.
\end{displaymath}
\begin{displaymath}
d_W \tilde cs_3 = \langle -,- \rangle
  \,.
\end{displaymath}
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 $0$.

\end{proof}
This is a special case of the more general statement below, about \href{OnShiftedCentralExtenstions}{invariant polynomials on shifted central extensions}.

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

\begin{defn}
\label{}\hypertarget{}{}
Write

\begin{displaymath}
(b \mathbb{R} \to \mathfrak{string}) \in L_\infty Alg
\end{displaymath}
for the [[L-∞ algebra]] defined by the fact that its [[Chevalley-Eilenberg algebra]] is given by

\begin{displaymath}
d_{CE} t^a = - \frac{1}{2} C^a{}_{b c} t^b \wedge t^c
\end{displaymath}
\begin{displaymath}
d_{CE} b = \mu_3 - c
\end{displaymath}
\begin{displaymath}
d_{CE} c = 0
  \,,
\end{displaymath}
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 cohomology|Lie algebra cocycle]] in degree 3, as above.

\end{defn}
It is easily seen that

\begin{prop}
\label{}\hypertarget{}{}
The canonical morphism

\begin{displaymath}
\mathfrak{g} \to (b \mathbb{R} \to \mathfrak{string})
\end{displaymath}
given dually by sending

\begin{displaymath}
t^a \mapsto t^a\,,\,\,\, b \mapsto 0\,,\, \,\, c \mapsto \mu_3
\end{displaymath}
is a weak equivalence.

\end{prop}
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 structure]]s, where the shifted piece $b \mathbb{R}$ in $(b \mathbb{R} \to \mathfrak{string})$ picks up the failure of $\mathfrak{so}$-valued [[connection on an ∞-bundle|connections]] to lift to $\mathfrak{string}$-[[connection on a 2-bundle|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 $\mathfrak{g}$.

\begin{prop}
\label{}\hypertarget{}{}
There is an [[isomorphism]]

\begin{displaymath}
inv(\mathfrak{g}) \simeq inv(b \mathbb{R} \to \mathfrak{string})
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
Notice that the [[Weil algebra]] of $(b\mathbb{R} \to \mathfrak{string})$ is given by

\begin{displaymath}
d_W t^a = - \frac{1}{2} C^a{}_{b c} t^b \wedge t^c + r^a
\end{displaymath}
\begin{displaymath}
d_W b = \mu_3 - c - h
\end{displaymath}
\begin{displaymath}
d_W c = g
\end{displaymath}
for new generators $\{r^a\}$ in degree 2, $h$ in degree 3 and $g$ in degree 4, coming with their [[Bianchi identities]]

\begin{displaymath}
d_W r^a = - C^a{}_{b c}t^b \wedge r^c
\end{displaymath}
\begin{displaymath}
d_W h = d_W \mu_3 - g
\end{displaymath}
\begin{displaymath}
d_W g = 0
\end{displaymath}
For the following computations let $\{k_{a b}\}$ be the structure constants of the [[Killing form]], so that

\begin{displaymath}
\langle -,- \rangle = k_{a b} r^a \wedge r^b
\end{displaymath}
and assume that $\mu_3$ is normalized such that

\begin{displaymath}
\mu_3 = k_{a a'}C^{a'}_{b c} t^a \wedge t^b \wedge t^c
\end{displaymath}
(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

\begin{displaymath}
\langle -,-\rangle = g + d_W (cs - \mu_3 + h)
 \,,
\end{displaymath}
where $cs$ is any [[Chern-Simons element]] for $\langle - , \rangle$, for instance

\begin{displaymath}
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
  \,,
\end{displaymath}
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 \emph{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.

\end{proof}
\hypertarget{on_symplectic_lie_algebroids}{}\subsubsection*{{On symplectic Lie $n$-algebroids}}\label{on_symplectic_lie_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 $n = 0$.

\hypertarget{Properties}{}\subsection*{{Properties}}\label{Properties}

\hypertarget{AsFormsOnBGconn}{}\subsubsection*{{As differential forms on the moduli stack of connections}}\label{AsFormsOnBGconn}

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]] \hyperlink{FreedHopkins13}{Freed-Hopkins 13}.

\begin{displaymath}
\mathrm{inv}(\mathfrak{g})\simeq \Omega^\bullet(\mathbf{B}G_{conn})
  \,.
\end{displaymath}
For more on this see also at \emph{\href{Weil+algebra#CharacterizationInSmoothTopos}{Weil algebra -- Characterization in the smooth infinity-topos}}.

\hypertarget{OnReductiveLieAlg}{}\subsubsection*{{On reductive Lie algebras}}\label{OnReductiveLieAlg}

\begin{prop}
\label{}\hypertarget{}{}
Let $\mathfrak{g}$ be a [[reductive Lie algebra]]. Then the subalgebra of invariant polynomials in the [[Weil algebra]] is the [[free construction|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]] $H^\bullet(\mathfrak{g}) = H^\bullet(CE(\mathfrak{g}))$.

\end{prop}
This appears for instance as (\hyperlink{GHV}{GHV, vol III, page 242, theorem I}).

\hypertarget{role_in_chernweil_theory}{}\subsection*{{Role in $\infty$-Chern-Weil theory}}\label{role_in_chernweil_theory}

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

\hypertarget{TransgressionCocycles}{}\subsubsection*{{Transgression cocycles and Chern-Simons elements}}\label{TransgressionCocycles}

\begin{udefn}
\textbf{(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

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

\begin{displaymath}
\mu := cs|_{\wedge^\bullet \mathfrak{g}^*} \in CE(\mathfrak{g})
\end{displaymath}
is a [[∞-Lie algebra cohomology|∞-Lie algebra cocycle]]. We say this is \emph{in transgression} with $\langle -\rangle$.

In total this construction yields a commuting diagram

\begin{displaymath}
\itexarray{
     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
  }
  \,,
\end{displaymath}
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 $b^{n-1} \mathbb{R}$.

\end{udefn}
\begin{uprop}
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.

\end{uprop}
\begin{proof}
The procedure that assigns $\mu$ to $\langle- \rangle$ is illustarted by the following diagram

\begin{displaymath}
\itexarray{
    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})    
  }
\end{displaymath}
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

\begin{itemize}%
\item the element $\mu$ satisfies $d_{CE(\mathfrak{a})} \mu = 0$, hence that it is an [[∞-Lie algebra cohomology|∞-Lie algebra cocycle]];


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



\end{itemize}
\end{proof}
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 $\mathfrak{a}$

\begin{displaymath}
inv(\mathfrak{a}) \subset CE(\Sigma \mathfrak{a}) = ker(W(\mathfrak{a}) \to CE(\mathfrak{a}))
  \,.
\end{displaymath}
From the short [[nLab:exact sequence]]

\begin{displaymath}
CE(\Sigma \mathfrak{a}) \to W(\mathfrak{a}) \to 
  CE(\mathfrak{a})
\end{displaymath}
we obtain the long exact sequence in [[chain homology and cohomology|cohomology]]

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

\begin{displaymath}
\delta : [\mu] \mapsto [\omega]
  \,.
\end{displaymath}
\textbf{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 \hyperlink{SemisimpLie}{On semisimple Lie algebras}.

\hypertarget{chernsimons_and_curvature_characteristic_forms}{}\subsubsection*{{Chern-Simons and curvature characteristic forms}}\label{chernsimons_and_curvature_characteristic_forms}

For $\mathfrak{g}$ a [[∞-Lie algebra|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 space|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

\begin{displaymath}
\mathbf{B}G = 
  \mathbf{cosk}_{n+1}((U,[k]) \mapsto \{ 
     \Omega^\bullet_{vert}(U \times \Delta^k)
     \leftarrow
     CE(\mathfrak{g})
  \})
  \,.
\end{displaymath}
We say an $\infty$-connection on this is an extension to a cocycle with coefficients in the simplicial sheaf

\begin{displaymath}
\mathbf{B}G_{diff} = 
  \mathbf{cosk}_{n+1}((U,[k]) \mapsto \left\{ 
    \itexarray{
       \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\}
  \,.
\end{displaymath}
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 \emph{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

\begin{displaymath}
\itexarray{
     \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
   }
  \,.
\end{displaymath}
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 $b^n \mathbb{R}$

\begin{displaymath}
\itexarray{
     \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
   }
  \,,
\end{displaymath}
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]].

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[∞-Lie algebra cocycle]]


\item [[Chern-Simons element]]


\item \textbf{invariant polynomial}, [[invariant theory]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

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

\begin{itemize}%
\item [[José de Azcárraga]], A. J. Macfarlane, A. J. Mountain, J. C. Perez Bueno, \emph{Invariant tensors for simple groups}, Nucl. Phys. B510 (1998) 657-687 (\href{http://arxiv.org/abs/physics/9706006}{arXiv:physics/9706006})

\end{itemize}
A standard textbook account of the traditional theory is in volume III of

\begin{itemize}%
\item [[Werner Greub]], [[Stephen Halperin]], [[Ray Vanstone]], \emph{[[Connections, Curvature, and Cohomology]]} Academic Press (1973)

\end{itemize}
The notion of invariant polynomials of $L_\infty$-algebras has been introduced in

\begin{itemize}%
\item [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]], \emph{$L_\infty$-connections} ().

\end{itemize}
The abstract characterization is due to

\begin{itemize}%
\item [[Daniel Freed]], [[Michael Hopkins]], \emph{Chern-Weil forms and abstract homotopy theory}, Bull. Amer. Math. Soc. 50 (2013), 431-468 (\href{http://arxiv.org/abs/1301.5959}{arXiv:1301.5959})

\end{itemize}
An account in the more general context of Lie theory in [[cohesive (infinity,1)-toposes]] is in section 3.3.11 of

\begin{itemize}%
\item [[Urs Schreiber]], \emph{[[schreiber:differential cohomology in a cohesive topos]]} .

\end{itemize}
[[!redirects algebra of invariant polynomials]] [[!redirects dg-algebra of invariant polynomials]]

[[!redirects invariant polynomials]]



\end{document}