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\section*{Chevalley-Eilenberg algebra}

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

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

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

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{GradingConvention}{Grading conventions}\dotfill \pageref*{GradingConvention} \linebreak
\noindent\hyperlink{OfLieAlgebra}{Of Lie algebras}\dotfill \pageref*{OfLieAlgebra} \linebreak
\noindent\hyperlink{DefForLieAlg}{Definition}\dotfill \pageref*{DefForLieAlg} \linebreak
\noindent\hyperlink{PropertiesForLieAlg}{Properties}\dotfill \pageref*{PropertiesForLieAlg} \linebreak
\noindent\hyperlink{of_algebras}{Of $L_\infty$-algebras}\dotfill \pageref*{of_algebras} \linebreak
\noindent\hyperlink{OfLieAlgebroids}{Of Lie algebroids}\dotfill \pageref*{OfLieAlgebroids} \linebreak
\noindent\hyperlink{of_lie_algebroids_2}{Of $\infty$-Lie algebroids}\dotfill \pageref*{of_lie_algebroids_2} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{of_abelian_lie_algebras}{Of abelian Lie $n$-algebras}\dotfill \pageref*{of_abelian_lie_algebras} \linebreak
\noindent\hyperlink{of_}{Of $\mathfrak{su}(2)$}\dotfill \pageref*{of_} \linebreak
\noindent\hyperlink{of_the_tangent_lie_algebroid_}{Of the tangent Lie algebroid $T X$}\dotfill \pageref*{of_the_tangent_lie_algebroid_} \linebreak
\noindent\hyperlink{of_the_string_lie_2algebra}{Of the string Lie 2-algebra}\dotfill \pageref*{of_the_string_lie_2algebra} \linebreak
\noindent\hyperlink{weil_algebra}{Weil algebra}\dotfill \pageref*{weil_algebra} \linebreak
\noindent\hyperlink{lie_algebra_cohomology}{Lie algebra cohomology}\dotfill \pageref*{lie_algebra_cohomology} \linebreak
\noindent\hyperlink{brst_complex}{BRST complex}\dotfill \pageref*{brst_complex} \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}

The \emph{Chevalley-Eilenberg algebra} $CE(\mathfrak{g})$ of a [[Lie algebra]] is a [[differential graded algebra]] of elements dual to $\mathfrak{g}$ whose [[differential]] encodes the Lie bracket on $\mathfrak{g}$.

The [[cochain cohomology]] of the underlying [[cochain complex]] is the [[Lie algebra cohomology]] of $\mathfrak{g}$.

This generalizes to a notion of Chevalley-Eilenberg algebra for $\mathfrak{g}$ an [[L-∞-algebra]], a [[Lie algebroid]] and generally an [[∞-Lie algebroid]].

\hypertarget{GradingConvention}{}\subsection*{{Grading conventions}}\label{GradingConvention}

This differential-graded subject is somewhat notorious for a plethora of equivalent but different conventions on gradings and signs.

For the following we adopt the convention that for $V$ an $\mathbb{N}$-[[graded vector space]] we write

\begin{displaymath}
\begin{aligned}
    \wedge^\bullet V &:= Sym(V[1])
     \\
      & = k \oplus (V_0) \oplus (V_1 \oplus V_0 \wedge V_0) \oplus 
     (V_2 \oplus V_1 \wedge V_0 \oplus V_0 \wedge V_0 \wedge V_0)
     \oplus \cdots
  \end{aligned}
\end{displaymath}
for the [[free construction|free]] graded-commutative algebra on the graded vector space obtained by shifting $V$ \emph{up} in degree by one.

Here the elements in the $n$th term in parenthesis are in degree $n$.

A plain vector space, such as the dual $\mathfrak{g}^*$ of the vector space underlying a Lie algebra, we regard her as a $\mathbb{N}$-graded vector space in degree 0. For such, $\wedge^\bullet \mathfrak{g}^*$ is the ordinary [[Grassmann algebra]] over $\mathfrak{g}^*$, where elements of $\mathfrak{g}^*$ are generators of degree 1.

\hypertarget{OfLieAlgebra}{}\subsection*{{Of Lie algebras}}\label{OfLieAlgebra}

\hypertarget{DefForLieAlg}{}\subsubsection*{{Definition}}\label{DefForLieAlg}

The \emph{Chevalley-Eilenberg algebra} $CE(\mathfrak{g})$ of a finite dimensional [[Lie algebra]] $\mathfrak{g}$ is the [[semifree dga|semifree]] graded-commutative [[dg-algebra]] whose underlying graded algebra is the [[Grassmann algebra]]

\begin{displaymath}
\wedge^\bullet \mathfrak{g}^*
  = 
  k \oplus \mathfrak{g}^* \oplus (\mathfrak{g}^* \wedge \mathfrak{g}^* ) \oplus \cdots
\end{displaymath}
(with the $n$th skew-symmetrized power in degree $n$)

and whose [[differential]] $d$ (of degree +1) is on $\mathfrak{g}^*$ the dual of the Lie bracket

\begin{displaymath}
d|_{\mathfrak{g}^*} := [-,-]^* : \mathfrak{g}^* \to \mathfrak{g}^* \wedge \mathfrak{g}^*
\end{displaymath}
extended uniquely as a graded [[derivation]] on $\wedge^\bullet \mathfrak{g}^*$.

That this differential indeed squares to 0, $d \circ d = 0$, is precisely the fact that the Lie bracket satisfies the [[Jacobi identity]].

If we choose a dual basis $\{t^a\}$ of $\mathfrak{g}^*$ and let $\{C^a{}_{b c}\}$ be the structure constants of the Lie bracket in that basis, then the action of the differential on the basis generators is

\begin{displaymath}
d t^a = - \frac{1}{2} C^a{}_{b c} t^b \wedge t^c
  \,,
\end{displaymath}
where here and in the following a sum over repeated indices is implicit.

This has a more or less evident generalization to infinite-dimensional Lie algebras.,

\hypertarget{PropertiesForLieAlg}{}\subsubsection*{{Properties}}\label{PropertiesForLieAlg}

One observes that for $\mathfrak{g}$ a vector space, the graded-commutative dg-algebra structures on $\wedge^\bullet \mathfrak{g}^*$ are precisely in bijection with Lie algebra structures on $\mathfrak{g}$: the dual of the restriction of $d$ to $\mathfrak{g}^*$ defines a skew-linear bracket and the condition $d^2 = 0$ holds if and only if that bracket satisfies the Jacobi identity.

Moreover, morphisms if Lie algebras $\mathfrak{g} \to \mathfrak{h}$ are precisely in bijection with morphisms of dg-algebras $CE(\mathfrak{g}) \leftarrow CE(\mathfrak{h})$. And the $CE$-construction is functorial.

Therefore, if we write $dgAlg_{sf,1}$ for the category whose objects are [[semifree dga]]s on generators in degree 1, we find that the construction of CE-algebras from Lie algebras constitutes a canonical [[equivalence of categories]]

\begin{displaymath}
LieAlg \stackrel{CE(-)}{\underset{\simeq}{\to}} (dgAlg_{sf,1})^{op}
  \,,
\end{displaymath}
where on the right we have the [[opposite category]].

This says that in a sense the Chevalley-Eilenberg algebra is just another way of looking at (finite dimensional) Lie algebras.

There is an analogous statement not involving the dualization: Lie algebra structures on $\mathfrak{g}$ are also in bijection with the structure of a \emph{[[differential graded coalgebra]]} $(\vee^\bullet \mathfrak{g}, D)$ on the free graded-co-commutative coalgebra $\vee^\bullet \mathfrak{g}$ on $\mathfrak{g}$ with $D$ a [[derivation]] of degree -1 squaring to 0.

The relation between the differentials is simply dualization

\begin{displaymath}
(\vee^\bullet \mathfrak{g}, D) \leftrightarrow
  (\wedge^\bullet \mathfrak{g}^* , d )
\end{displaymath}
where for each $\omega \in \wedge^\bullet \mathfrak{g}^*$ we have

\begin{displaymath}
d \omega = \omega(D(-))
  \,.
\end{displaymath}
\hypertarget{of_algebras}{}\subsection*{{Of $L_\infty$-algebras}}\label{of_algebras}

The equivalence between Lie algebras and differential graded algebras/coalgebras discussed above suggests a grand generalization by simply generalizing the [[Grassmann algebra]] over a [[vector space]] $\mathfrak{g}^*$ to the Grassmann algebra over a [[graded vector space]].

If $\mathfrak{g}$ is a graded vector space, then a differential $D$ of degree -1 squaring to 0 on $\vee^\bullet \mathfrak{g}$ is precisely the same as equipping $\mathfrak{g}$ with the structure of an [[L-∞ algebra]].

Dually, this corresponds to a general [[semifree dga]]

\begin{displaymath}
CE(\mathfrak{g}) := (\wedge^\bullet \mathfrak{g}^*, d = D^*)
  \,.
\end{displaymath}
This we may usefully think of as the Chevalley-Eilenberg algebra of the $L_\infty$-algebra $\mathfrak{g}$.

So \emph{every} commutative [[semifree dga]] (degreewise finite-dimensional) is the Chevaley-Eilenberg algebra of some [[L-∞ algebra]] of finite type.

This means that many constructions involving [[dg-algebra]]s are secretly about [[∞-Lie theory]]. For instance the [[Sullivan construction]] in [[rational homotopy theory]] may be interpreted in terms of [[Lie integration]] of $L_\infty$-algebras.

\hypertarget{OfLieAlgebroids}{}\subsection*{{Of Lie algebroids}}\label{OfLieAlgebroids}

For $\mathfrak{a}$ a [[Lie algebroid]] given as

\begin{itemize}%
\item a [[vector bundle]] $E\to X$


\item with anchor map $\rho : E \to T X$


\item and bracket $[-,-] \;\colon\; \Gamma(E)\wedge_{\mathbb{R}} \Gamma(E) \to \Gamma(E)$



\end{itemize}
the corresponding Chevalley-Eilenberg algebra is

\begin{displaymath}
CE(\mathfrak{a}) := \left(\wedge^\bullet_{C^\infty(X)} \Gamma(E)^*, d\right)
  \,,
\end{displaymath}
where now the [[tensor product]]s and dualization is over the ring $C^\infty(X)$ of [[smooth function]]s on the base space $X$ (with values in the [[real number]]s). The differential $d$ is given by the formula

\begin{displaymath}
(d\omega)(e_0, \cdots, e_n)
  = 
  \sum_{\sigma \in Shuff(1,n)} sgn(\sigma) \rho(e_{\sigma(0)})(\omega(e_{\sigma(1)}, \cdots, e_{\sigma(n)}))
  +
  \sum_{\sigma \in Shuff(2,n-1)} sign(\sigma) 
   \omega([e_{\sigma(0)},e_{\sigma(1)}],e_{\sigma(2)}, \cdots, e_{\sigma(n)} )
  \,,
\end{displaymath}
for all $\omega \in \wedge^n_{C^\infty(X)} \Gamma(E)^*$ and $(e_i \in \Gamma(E))$, where $Shuff(p,q)$ denotes the set of $(p,q)$-[[shuffle]]s $\sigma$ and $sgn(\sigma)$ the [[signature]] $\in \{\pm 1\}$ of the corresponding [[permutation]].

For $X = *$ the point we have that $\mathfrak{a}$ is a Lie algebra and this definition reproduces the above definition of the CE-algebra of a Lie algebra (possibly up to an irrelevant global sign).

\hypertarget{of_lie_algebroids_2}{}\subsection*{{Of $\infty$-Lie algebroids}}\label{of_lie_algebroids_2}

See [[∞-Lie algebroid]].

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{of_abelian_lie_algebras}{}\subsubsection*{{Of abelian Lie $n$-algebras}}\label{of_abelian_lie_algebras}

The CE-algebra of the Lie algebra of the [[circle group]] $\mathfrak{u}(1)$ is the graded-commutative dg-algebra on a single generator in degree 1 with vanishing differential.

More generally, the $L_\infty$-algebra $b^n \mathfrak{u}(1)$ is the one whose CE algebra is the commutative dg-algebra with a single generator in degree $n+1$ and vanishing differential.

\hypertarget{of_}{}\subsubsection*{{Of $\mathfrak{su}(2)$}}\label{of_}

The CE-algebra of $\mathfrak{su}(2)$ has three generators $x, y, z$ in degree one and differential

\begin{displaymath}
d x_1 = x_2 \wedge x_3
\end{displaymath}
and cyclically.

\hypertarget{of_the_tangent_lie_algebroid_}{}\subsubsection*{{Of the tangent Lie algebroid $T X$}}\label{of_the_tangent_lie_algebroid_}

For $X$ a [[smooth manifold]] and $T X$ its [[tangent Lie algebroid]], the corresponding CE-algebra is the [[de Rham algebra]] of $X$.

\begin{displaymath}
CE(T X) = (\wedge^\bullet_{C^\infty(X)} \Gamma(T^* X), d_{dR})
  \,.
\end{displaymath}
For $(v_i \in \Gamma(T X))$ [[vector field]]s and $\omega \in \Omega^n = \wedge^n_{C^\infty(X)} \Gamma(T X)^*$ a [[differential form]] of degree $n$, the \hyperlink{OfLieAlgebroids}{formula for the CE-differential}

\begin{displaymath}
(d\omega)(v_0, \cdots, v_n)
  = 
  \sum_{\sigma \in Sh(1,n)} sgn(\sigma) v_{\sigma(0)}(\omega(v_{\sigma(1)}, \cdots, v_{\sigma(n)}))
  +
  \sum_{\sigma \in Shuff(2,n-1)} sgn(\sigma) 
   \omega([v_{\sigma(0)},v_{\sigma(1)}],v_{\sigma(2)}, \cdots, v_{\sigma(n)} )
  \,,
\end{displaymath}
is indeed that for the de Rham differential.

\hypertarget{of_the_string_lie_2algebra}{}\subsubsection*{{Of the string Lie 2-algebra}}\label{of_the_string_lie_2algebra}

For $\mathfrak{g}$ a semisimple Lie algebra with binary [[invariant polynomial]] $\langle -,-\rangle$ -- the [[Killing form]] -- , the CE-algebra of the [[string Lie 2-algebra]] is

\begin{displaymath}
CE(\mathfrak{string}) = (\wedge^\bullet( \mathfrak{g}^+ \oplus \mathbb{R}^*[1]), d_{string})
\end{displaymath}
where the differential restricted to $\mathfrak{g}^*$ is $[-,-]^*$ while on the new generator $b$ spanning $\mathbb{R}^*[1]$ it is

\begin{displaymath}
d b = \langle -, [-,-]\rangle \in \wedge^3 \mathfrak{g}^*
  \,.
\end{displaymath}
\hypertarget{weil_algebra}{}\subsubsection*{{Weil algebra}}\label{weil_algebra}

For $\mathfrak{g}$ a [[Lie algebra]], the CE-algebra of the [[Lie 2-algebra]] given by the [[differential crossed module]] $(\mathfrak{g} \stackrel{Id}{\to} \mathfrak{g})$ is the [[Weil algebra]] $W(\mathfrak{g})$ of $\mathfrak{g}$

\begin{displaymath}
CE(\mathfrak{g} \stackrel{Id}{\to} \mathfrak{g})
  =
  W(\mathfrak{g})
  \,.
\end{displaymath}
\hypertarget{lie_algebra_cohomology}{}\subsubsection*{{Lie algebra cohomology}}\label{lie_algebra_cohomology}

[[Lie algebra cohomology]] of a $k$-[[Lie algebra]] $\mathfrak{g}$ with coefficients in the left $\mathfrak{g}$-module $M$ is defined as $H^*_{Lie}(\mathfrak{g},M) = Ext_{U\mathfrak{g}}^*(k,M)$. It can be computed as $Hom_{\mathfrak{g}}(V(\mathfrak{g}),M)$ (a similar story is for [[Lie algebra homology]]) where $V(\mathfrak{g})=U(\mathfrak{g})\otimes\Lambda^*(\mathfrak{g})$ is the [[Chevalley-Eilenberg chain complex]]. If $\mathfrak{g}$ is finite-dimensional over a field then $Hom_{\mathfrak{g}}(V(\mathfrak{g}),k) =  CE(\mathfrak{g}) = \Lambda^* \mathfrak{g}^*$ is the underlying complex of the Chevalley-Eilenberg algebra, i.e. the [[Chevalley-Eilenberg cochain complex]] with trivial coefficients.

A [[cocycle]] in degree n of the [[Lie algebra cohomology]] of a [[Lie algebra]] $\mathfrak{g}$ with values in the trivial module $\mathbb{R}$ is a morphism of [[L-∞ algebra]]s

\begin{displaymath}
\mathfrak{g} \to b^{n-1} \mathfrak{u}(1)
  \,.
\end{displaymath}
In terms of CE-algebras this is a [[dg-algebra]] morphism

\begin{displaymath}
CE(\mathfrak{g}) \leftarrow CE(b^{n-1}\mathfrak{u}(1))
  \,.
\end{displaymath}
Since by the above example the dg-algebra on he right has a single generator in degree $n$ and vanishing differential, such a morphism is precisely the same thing as a degree $n$-element in $CE(\mathfrak{g})$, i.e. an element $\omega \in \wedge^n \mathfrak{g}^*$ which is closed under the CE-differential

\begin{displaymath}
d_{CE} \omega = 0
  \,.
\end{displaymath}
This is what one often sees as the definition of a cocycle in [[Lie algebra cohomology]]. However, from the general point of view of [[cohomology]], it is better to think of the cocycle equivalently as the morphism $\mathfrak{g} \to b^{n-1}\mathfrak{u}(1)$.

\hypertarget{brst_complex}{}\subsubsection*{{BRST complex}}\label{brst_complex}

In [[physics]], the Chevalley-Eilenberg algebra $CE(\mathfrak{g}, N)$ of the [[action]] of a [[Lie algebra]] or [[L-∞ algebra]] of a [[gauge group]] $G$ on space $N$ of fields is called the [[BRST complex]].

In this context

\begin{itemize}%
\item the generators in $N$ in degree 0 are called \textbf{fields};


\item the generators $\in \mathfrak{g}^*$ in degree $1$ are called \textbf{ghosts};


\item the generators in degree $2$ are called \textbf{ghosts of ghosts};


\item etc.



\end{itemize}
If $N$ is itself a [[chain complex]], then this is called a [[BV-BRST formalism|BV-BRST complex]]

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

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


\item [[super L-∞ algebra]]


\item \textbf{Chevalley-Eilenberg algebra}

\begin{itemize}%
\item [[differential graded-commutative algebra]]


\item [[differential graded-commutative superalgebra]]



\end{itemize}

\item [[Weil algebra]]


\item [[invariant polynomial]]



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

An elementary introduction for CE-algebras of Lie algebras is at the beginning of

\begin{itemize}%
\item [[José de Azcárraga]], J. M. Izquierdo, J. C. Perez Bueno, \emph{An introduction to some novel applications of Lie algebra cohomology and physics} (\href{http://arxiv.org/abs/physics/9803046}{arXiv:physics/9803046})

\end{itemize}
More details are in of

\begin{itemize}%
\item [[José de Azcárraga]], Jos\'e{} M. Izquierdo, section 6.7 of \emph{[[Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics]]} , Cambridge monographs of mathematical physics, (1995)

\end{itemize}
See also almost any text on [[Lie algebra cohomology]] (see the list of references there).

[[!redirects Chevalley--Eilenberg algebra]] [[!redirects Chevalley–Eilenberg algebra]] [[!redirects Chevalley-Eilenberg algebras]] [[!redirects Chevalley--Eilenberg algebras]] [[!redirects Chevalley–Eilenberg algebras]]

[[!redirects Chevalley-Eilenberg dg-algebra]] [[!redirects Chevalley-Eilenberg dg-algebras]]

[[!redirects Chevalley-Eilenberg differential]] [[!redirects Chevalley-Eilenberg differentials]]



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