\documentclass[12pt,titlepage]{article} \usepackage{amsmath} \usepackage{mathrsfs} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsthm} \usepackage{mathtools} \usepackage{graphicx} \usepackage{color} \usepackage{ucs} \usepackage[utf8x]{inputenc} \usepackage{xparse} \usepackage{hyperref} %----Macros---------- % % Unresolved issues: % % \righttoleftarrow % \lefttorightarrow % % \color{} with HTML colorspec % \bgcolor % \array with options (without options, it's equivalent to the matrix environment) % Of the standard HTML named colors, white, black, red, green, blue and yellow % are predefined in the color package. Here are the rest. \definecolor{aqua}{rgb}{0, 1.0, 1.0} \definecolor{fuschia}{rgb}{1.0, 0, 1.0} \definecolor{gray}{rgb}{0.502, 0.502, 0.502} \definecolor{lime}{rgb}{0, 1.0, 0} \definecolor{maroon}{rgb}{0.502, 0, 0} \definecolor{navy}{rgb}{0, 0, 0.502} \definecolor{olive}{rgb}{0.502, 0.502, 0} \definecolor{purple}{rgb}{0.502, 0, 0.502} \definecolor{silver}{rgb}{0.753, 0.753, 0.753} \definecolor{teal}{rgb}{0, 0.502, 0.502} % Because of conflicts, \space and \mathop are converted to % \itexspace and \operatorname during preprocessing. % itex: \space{ht}{dp}{wd} % % Height and baseline depth measurements are in units of tenths of an ex while % the width is measured in tenths of an em. \makeatletter \newdimen\itex@wd% \newdimen\itex@dp% \newdimen\itex@thd% \def\itexspace#1#2#3{\itex@wd=#3em% \itex@wd=0.1\itex@wd% \itex@dp=#2ex% \itex@dp=0.1\itex@dp% \itex@thd=#1ex% \itex@thd=0.1\itex@thd% \advance\itex@thd\the\itex@dp% \makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}} \makeatother % \tensor and \multiscript \makeatletter \newif\if@sup \newtoks\@sups \def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}% \def\reset@sup{\@supfalse\@sups={}}% \def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else% \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}% \else \append@sup#2 \@suptrue \fi% \expandafter\mk@scripts\fi} \def\tensor#1#2{\reset@sup#1\mk@scripts#2_/} \def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2% \reset@sup\mk@scripts#3_/} \makeatother % \slash \makeatletter \newbox\slashbox \setbox\slashbox=\hbox{$/$} \def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$} \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa \copy\slashbox \kern-\@tempdima \box\@tempboxa} \def\slash{\protect\itex@pslash} \makeatother % math-mode versions of \rlap, etc % from Alexander Perlis, "A complement to \smash, \llap, and lap" % http://math.arizona.edu/~aprl/publications/mathclap/ \def\clap#1{\hbox to 0pt{\hss#1\hss}} \def\mathllap{\mathpalette\mathllapinternal} \def\mathrlap{\mathpalette\mathrlapinternal} \def\mathclap{\mathpalette\mathclapinternal} \def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}} \def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}} \def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}} % Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2} \let\oldroot\root \def\root#1#2{\oldroot #1 \of{#2}} \renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}} % Manually declare the txfonts symbolsC font \DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n} \SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n} \DeclareFontSubstitution{U}{txsyc}{m}{n} % Manually declare the stmaryrd font \DeclareSymbolFont{stmry}{U}{stmry}{m}{n} \SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n} % Manually declare the MnSymbolE font \DeclareFontFamily{OMX}{MnSymbolE}{} \DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n} \SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n} \DeclareFontShape{OMX}{MnSymbolE}{m}{n}{ <-6> MnSymbolE5 <6-7> MnSymbolE6 <7-8> MnSymbolE7 <8-9> MnSymbolE8 <9-10> MnSymbolE9 <10-12> MnSymbolE10 <12-> MnSymbolE12}{} % Declare specific arrows from txfonts without loading the full package \makeatletter \def\re@DeclareMathSymbol#1#2#3#4{% \let#1=\undefined \DeclareMathSymbol{#1}{#2}{#3}{#4}} \re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116} \re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116} \re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117} \re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117} \re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118} \re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118} \re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119} \re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119} \re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46} \re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121} \re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121} \re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12} \re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64} \re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6} \re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77} \re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77} \makeatother % \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE \makeatletter \def\Decl@Mn@Delim#1#2#3#4{% \if\relax\noexpand#1% \let#1\undefined \fi \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}} \def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}} \def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}} \Decl@Mn@Open{\llangle}{mnomx}{'164} \Decl@Mn@Close{\rrangle}{mnomx}{'171} \Decl@Mn@Open{\lmoustache}{mnomx}{'245} \Decl@Mn@Close{\rmoustache}{mnomx}{'244} \makeatother % Widecheck \makeatletter \DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}} \def\@widecheck#1#2{% \setbox\z@\hbox{\m@th$#1#2$}% \setbox\tw@\hbox{\m@th$#1% \widehat{% \vrule\@width\z@\@height\ht\z@ \vrule\@height\z@\@width\wd\z@}$}% \dp\tw@-\ht\z@ \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@ \setbox\tw@\hbox{% \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box \tw@}}}% {\ooalign{\box\tw@ \cr \box\z@}}} \makeatother % \mathraisebox{voffset}[height][depth]{something} \makeatletter \NewDocumentCommand\mathraisebox{moom}{% \IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{% \IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}% }{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}% \mathpalette\@temp{#4}} \makeatletter % udots (taken from yhmath) \makeatletter \def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.} \mkern2mu\raise4\p@\hbox{.}\mkern1mu \raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}} \makeatother %% Fix array \newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}} %% \itexnum is a noop \newcommand{\itexnum}[1]{#1} %% Renaming existing commands \newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}} \newcommand{\widevec}{\overrightarrow} \newcommand{\darr}{\downarrow} \newcommand{\nearr}{\nearrow} \newcommand{\nwarr}{\nwarrow} \newcommand{\searr}{\searrow} \newcommand{\swarr}{\swarrow} \newcommand{\curvearrowbotright}{\curvearrowright} \newcommand{\uparr}{\uparrow} \newcommand{\downuparrow}{\updownarrow} \newcommand{\duparr}{\updownarrow} \newcommand{\updarr}{\updownarrow} \newcommand{\gt}{>} \newcommand{\lt}{<} \newcommand{\map}{\mapsto} \newcommand{\embedsin}{\hookrightarrow} \newcommand{\Alpha}{A} \newcommand{\Beta}{B} \newcommand{\Zeta}{Z} \newcommand{\Eta}{H} \newcommand{\Iota}{I} \newcommand{\Kappa}{K} \newcommand{\Mu}{M} \newcommand{\Nu}{N} \newcommand{\Rho}{P} \newcommand{\Tau}{T} \newcommand{\Upsi}{\Upsilon} \newcommand{\omicron}{o} \newcommand{\lang}{\langle} \newcommand{\rang}{\rangle} \newcommand{\Union}{\bigcup} \newcommand{\Intersection}{\bigcap} \newcommand{\Oplus}{\bigoplus} \newcommand{\Otimes}{\bigotimes} \newcommand{\Wedge}{\bigwedge} \newcommand{\Vee}{\bigvee} \newcommand{\coproduct}{\coprod} \newcommand{\product}{\prod} \newcommand{\closure}{\overline} \newcommand{\integral}{\int} \newcommand{\doubleintegral}{\iint} \newcommand{\tripleintegral}{\iiint} \newcommand{\quadrupleintegral}{\iiiint} \newcommand{\conint}{\oint} \newcommand{\contourintegral}{\oint} \newcommand{\infinity}{\infty} \newcommand{\bottom}{\bot} \newcommand{\minusb}{\boxminus} \newcommand{\plusb}{\boxplus} \newcommand{\timesb}{\boxtimes} \newcommand{\intersection}{\cap} \newcommand{\union}{\cup} \newcommand{\Del}{\nabla} \newcommand{\odash}{\circleddash} \newcommand{\negspace}{\!} \newcommand{\widebar}{\overline} \newcommand{\textsize}{\normalsize} \renewcommand{\scriptsize}{\scriptstyle} \newcommand{\scriptscriptsize}{\scriptscriptstyle} \newcommand{\mathfr}{\mathfrak} \newcommand{\statusline}[2]{#2} \newcommand{\tooltip}[2]{#2} \newcommand{\toggle}[2]{#2} % Theorem Environments \theoremstyle{plain} \newtheorem{theorem}{Theorem} \newtheorem{lemma}{Lemma} \newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{Weil algebra} \begin{quote}% There are two different concepts called \emph{Weil algebra}. This entry is about the notion of Weil algebra in [[Lie theory]]. For the notion in [[infinitesimal object|infinitesimal]] geometry see [[infinitesimally thickened point]]/[[Artin algebra]]. \end{quote} \vspace{.5em} \hrule \vspace{.5em} \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{weil_algebra_of_a_lie_algebra}{Weil algebra of a Lie algebra}\dotfill \pageref*{weil_algebra_of_a_lie_algebra} \linebreak \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{for_algebras}{For $L_\infty$-algebras}\dotfill \pageref*{for_algebras} \linebreak \noindent\hyperlink{for_algebroids}{For $L_\infty$-algebroids}\dotfill \pageref*{for_algebroids} \linebreak \noindent\hyperlink{Properties}{Properties}\dotfill \pageref*{Properties} \linebreak \noindent\hyperlink{FreeProperty}{Free property}\dotfill \pageref*{FreeProperty} \linebreak \noindent\hyperlink{CharacterizationInSmoothTopos}{Characterization in the smooth $\infty$-topos}\dotfill \pageref*{CharacterizationInSmoothTopos} \linebreak \noindent\hyperlink{relation_to_cartan_model_for_equivariant_de_rham_cohomology}{Relation to Cartan model for equivariant de Rham cohomology}\dotfill \pageref*{relation_to_cartan_model_for_equivariant_de_rham_cohomology} \linebreak \noindent\hyperlink{AsInnerDer}{As the CE-algebra of the $L_\infty$-algebra of inner derivations}\dotfill \pageref*{AsInnerDer} \linebreak \noindent\hyperlink{for_an_ordinary_lie_algebra}{For an ordinary Lie algebra}\dotfill \pageref*{for_an_ordinary_lie_algebra} \linebreak \noindent\hyperlink{relation_to_other_concepts}{Relation to other concepts}\dotfill \pageref*{relation_to_other_concepts} \linebreak \noindent\hyperlink{LieAlgValuedForms}{$\infty$-Lie algebra valued differential forms}\dotfill \pageref*{LieAlgValuedForms} \linebreak \noindent\hyperlink{invariant_polynomials_and_chernsimons_elements}{Invariant polynomials and Chern-Simons elements}\dotfill \pageref*{invariant_polynomials_and_chernsimons_elements} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{WeilofLieAlg}{Weil algebra of a Lie algebra}\dotfill \pageref*{WeilofLieAlg} \linebreak \noindent\hyperlink{weil_algebra_of_a_0lie_algebroid}{Weil algebra of a 0-Lie algebroid}\dotfill \pageref*{weil_algebra_of_a_0lie_algebroid} \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 notion of \emph{Weil algebra} is ordinarily defined for a [[Lie algebra]] $\mathfrak{g}$. It may be understood as the [[Chevalley-Eilenberg algebra]] of the tangent [[Lie 2-algebra]] $T \mathfrak{g}$ or $inn(\mathfrak{g})$ of $\mathfrak{g}$, generalizing the notion of [[tangent Lie algebroid]] $T X$ from a 0-[[truncated]] Lie algebroid $X$ (a [[smooth manifold]]) to the one-obeject [[Lie algebroid]] $\mathfrak{g}$. Generally, for every [[Lie-∞-algebroid]] $\mathfrak{a}$ one may define the corresponding tangent Lie-$\infty$-algebroid $T \mathfrak{a}$, whose Chevalley-Eilenberg algebra may be called the Weil algebra of $\mathfrak{a}$: \begin{displaymath} W(\mathfrak{a}) = CE(T \mathfrak{a}) \,. \end{displaymath} \hypertarget{weil_algebra_of_a_lie_algebra}{}\subsubsection*{{Weil algebra of a Lie algebra}}\label{weil_algebra_of_a_lie_algebra} Let $\mathfrak{g}$ be a finite-dimensional [[Lie algebra]]. The \textbf{Weil algebra} $W(\mathfrak{g})$ of $\mathfrak{g}$ is \begin{itemize}% \item the graded [[Grassmann algebra]] generated from the dual [[vector space]] $\mathfrak{g}^*$ together with another copy of $\mathfrak{g}^*$ shifted in degree \begin{displaymath} W(\mathfrak{g}) := \wedge^\bullet (\mathfrak{g}^* \oplus \mathfrak{g}^*[1]) \end{displaymath} \item equipped with a [[derivation]] $d : W(\mathfrak{g}) \to W(\mathfrak{g})$ that makes this a [[dg-algebra]], defined by the fact that on $\mathfrak{g}^*$ it acts as the differential of the [[Chevalley-Eilenberg algebra]] of $\mathfrak{g}$ plus the degree shift morphism $\mathfrak{g}^* \to \mathfrak{g}^*$. \end{itemize} This Weil algebra has trivial [[chain homology and cohomology|cohomology]] everywhere (except in degree 0 of course) and sits in a sequence \begin{displaymath} CE(\mathfrak{g}) \leftarrow W(\mathfrak{g}) \leftarrow inv(\mathfrak{g}) \end{displaymath} with the [[Chevalley-Eilenberg algebra]] of $\mathfrak{g}$ and its algebra of [[invariant polynomial]]s on $\mathfrak{g}$. This may be understood as a model for the sequence of algebras of differential forms on the [[generalized universal bundle|universal G-bundle]] \begin{displaymath} G \to \mathcal{E}G \to \mathcal{B}G \,. \end{displaymath} As such, the Weil algebra plays a crucial role in the study of the [[Lie algebra cohomology]] of $\mathfrak{g}$. \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} We first consider Weil algebras of [[L-∞ algebra]]s, then more generally of [[L-∞ algebroid]]s. We use the notation and grading conventions that are described in detail at [[Chevalley-Eilenberg algebra]]. \hypertarget{for_algebras}{}\subsubsection*{{For $L_\infty$-algebras}}\label{for_algebras} Let $\mathfrak{g}$ be an [[L-∞ algebra]] of [[finite type]]. By our grading conventions this means that the [[graded vector space]] $\mathfrak{g}^*$ obtained by degreewise dualization is in non-negative degree, and $\wedge^1 \mathfrak{g}^* = \mathfrak{g}^*[1]$ is its shift up into positive degree. A quick abstract way to characterize the Weil algebra of $\mathfrak{g}$ is as follows. Notice that there is a [[free functor]]/[[forgetful functor]] [[adjunction]] \begin{displaymath} (F \dashv U) : dgAlg \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}} Vect[\mathbb{Z}] \end{displaymath} between the [[category]] [[dgAlg]] of [[dg-algebra]]s and the category of $\mathbb{Z}$-graded [[vector space]]s (all over some fixed [[field]]). Notice that a free object is unique \emph{up to [[isomorphism]]} . \begin{defn} \label{}\hypertarget{}{} The \textbf{Weil algebra} $W(\mathfrak{g})$ is the unique representative of the [[free functor|free]] [[dg-algebra]] on $\wedge^1 \mathfrak{g}^*$ for which the projection of graded vector spaces $\wedge^1(\mathfrak{g}^* \oplus \mathfrak{g}^*[1]) \to \wedge^1 \mathfrak{g}^*$ extended to a [[dg-algebra]] [[homomorphism]] $W(\mathfrak{g}) \to CE(\mathfrak{g})$ \end{defn} We discuss below in the \hyperlink{Properties}{Properties} section that this is equivalent to the following component-wise definition \begin{defn} \label{WeilForLInfinitityAlgebra}\hypertarget{WeilForLInfinitityAlgebra}{} The \textbf{Weil algebra} $W(\mathfrak{g})$ is the [[semi-free dga]] whose underlying graded-commutative algebra is the [[exterior algebra]] \begin{displaymath} \wedge^\bullet (\mathfrak{g}^* \oplus \mathfrak{g}^*[1]) \end{displaymath} on $\mathfrak{g}^*$ and a shifted copy of $\mathfrak{g}^*$, and whose [[differential]] is the sum \begin{displaymath} d_{W(\mathfrak{g})} = d_{CE(\mathfrak{g})} + \mathbf{d} \end{displaymath} of two graded [[derivations]] of degree +1 defined by \begin{itemize}% \item $\mathbf{d}$ acts by degree shift $\mathfrak{g}^* \to \mathfrak{g}^*[1]$ on elements in $\mathfrak{g}^*$ and by 0 on elements of $\mathfrak{g}^*[1]$; \item $d_{CE(\mathfrak{g})}$ acts on unshifted elements in $\mathfrak{g}^*$ as the differential of the [[Chevalley-Eilenberg algebra]] of $\mathfrak{g}$ and is extended uniquely to shifted generators by graded-commutativity \begin{displaymath} [d_{CE(\mathfrak{g})}, \mathbf{d}] = 0 \end{displaymath} with $\mathbf{d}$: \begin{displaymath} d_{CE(\mathfrak{g})} \mathbf{d} \omega := - \mathbf{d} d_{CE(\mathfrak{g})} \omega \end{displaymath} for all $\omega \in \wedge^1 \mathfrak{g}^*$. \end{itemize} \end{defn} \hypertarget{for_algebroids}{}\subsubsection*{{For $L_\infty$-algebroids}}\label{for_algebroids} Where the [[Chevalley-Eilenberg algebra]] of an [[L-∞ algebra]] has in degree 0 the ground field, that of an [[L-∞ algebroid]] has more generally an [[algebra over a Lawvere theory|algebra over]] a [[Lawvere theory]]. For [[L-∞ algebroid]]s over [[smooth manifold]]s this is the algebra of [[smooth function]]s on a manifolds, regarded as a [[smooth algebra]] ($C^\infty$-ring). So let $T$ be a [[Fermat theory]]. Write $T Alg$ for the corresponding [[category]] of [[algebra over a Lawvere theory|algebra]]. There is a [[free functor]]/[[forgetful functor]] [[adjunction]] \begin{displaymath} (F \dashv U) : T Alg \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}} CRing \end{displaymath} to the category [[CRing]] of commutative [[Ring]]s. We need the facts that \begin{itemize}% \item a [[module]] over a $T$-algebra $A$ is uniquely specified by its underlying module over $U(A)$; \item the universal [[derivation]] on a $T$-algebra $A$ is the [[de Rham differential]] \begin{displaymath} d_{dR} : A \to \Omega^1(A) \end{displaymath} \end{itemize} with values in the $A$-module of $T$-[[Kähler differential]]s. See the corresponding entries for more details. The second point means that for $v : A \to N$ any $T$-[[derivation]] on $A$, there is a unique $A$-[[module]] [[homomorphism]] \begin{displaymath} \Omega^\bullet(A) \to N \end{displaymath} such that the diagram \begin{displaymath} \itexarray{ && \Omega^\bullet(A) \\ & {}^{\mathllap{d_{dR}}}\nearrow & \downarrow^{\mathrlap{v}} \\ A &\stackrel{v}{\to}& N } \end{displaymath} commutes. Let now $\mathfrak{a}$ be an [[L-∞ algebroid]] with [[Chevalley-Eilenberg algebra]] considered as the following data; \begin{enumerate}% \item a graded commutative [[semifree dga]] $CE(\mathfrak{a})$ over the ground field; \item the structure of a $T$-[[algebra over a Lawvere theory|algebra]] on the [[associative algebra]] $A := CE(\mathfrak{a})_0$ (over the ground field) such that $d_{CE(\mathfrak{a})} : CE(\mathfrak{a})_0 \to CE(\mathfrak{a})_1$ is a [[derivation]] of $T$-algebra modules. \end{enumerate} By [[semifree dga|semi-freeness]] there exists a $\mathbb{N}$-[[graded vector space]] $(\mathfrak{a}^*)^\bullet$ and an [[isomorphism]] \begin{displaymath} CE(\mathfrak{a}) \simeq (\wedge^\bullet_{A} (\mathfrak{a}^*), d_{CE(\mathfrak{a})}) \,. \end{displaymath} \begin{defn} \label{}\hypertarget{}{} The \textbf{Weil algebra} $W(\mathfrak{a})$ of the $L_\infty$-algebroid $\mathfrak{a}$ is the Chevalley-Eilenberg algebra of the $L_\infty$-algebroid defined as follows \begin{itemize}% \item the $T$-algebra $A$ in degree 0 is the same as that of $\mathfrak{A}$; \item the underlying graded algebra is the [[exterior algebra]] on $\mathfrak{a}^*$ and a shifted copy $\mathfrak{a}^*[1]$ as well as one copy of the [[Kähler differential]] module $\Omega^1$ in lowest degree (though of as the shifted copy of $A$ itself) \begin{displaymath} \wedge^\bullet (\Omega^1(A) \oplus (\mathfrak{a}^*) \oplus \mathfrak{a}^*[1]) \,. \end{displaymath} \item the [[differential]] is the sum \begin{displaymath} d_{W(\mathfrak{a})} = d_{CE(\mathfrak{a})} + \mathbf{d} \end{displaymath} of two degree +1 graded derivations, where $d_{CE(\mathfrak{a})}$ and $\mathbf{a}$ are defined on $\wedge^1 \mathfrak{a}^* \oplus \mathfrak{a}^*[1]$ as \hyperlink{WeilForLInfinitityAlgebra}{above} for $L_\infty$-algebras and on $A$ itself $d_{CE(\mathfrak{a})}$ vanishes and $\mathbf{d}$ acts as the universal derivation \begin{displaymath} \mathbf{d}|_A = d_{\mathrm{dR}} : A \to \Omega^1(A) \,. \end{displaymath} \end{itemize} \end{defn} \hypertarget{Properties}{}\subsection*{{Properties}}\label{Properties} \hypertarget{FreeProperty}{}\subsubsection*{{Free property}}\label{FreeProperty} The main point of the definition is that the differential restricted to the original (unshifted) generators is the original differential plus the shift: \begin{displaymath} d_{W(\mathfrak{a})} |_{\mathfrak{a}^*} = d_{CE(\mathfrak{a})} + \mathbf{d} \,. \end{displaymath} By solving the condition $d_{W(\mathfrak{a})} \circ d_{W(\mathfrak{a})} = 0$ and using that $d_{CE(\mathfrak{a})} d_{CE(\mathfrak{a})} = 0$ this already fixes uniquely the differential $d_{W(\mathfrak{a})}$. To see this we only need to show that the value of $d_{W(\mathfrak{a})}(x)$ on a generator $x=\sigma(t) \in \mathfrak{a}^*[1]$ is completely determined by $d_{W(\mathfrak{a})}\vert_{\wedge^\bullet\mathfrak{a}^*}$. One computes: \begin{displaymath} \begin{aligned} 0 & = d_{W(\mathfrak{a})}(d_{W(\mathfrak{a})} t) \\ & = d_{W(\mathfrak{a})}(d_{CE(\mathfrak{a})}t + \sigma t) \\ & = \sigma d_{CE(\mathfrak{a})} t + d_{W(\mathfrak{a}) } x \end{aligned} \end{displaymath} and hence \begin{displaymath} d_{W(\mathfrak{a})} x = - \sigma d_{CE(\mathfrak{a})} \sigma^{-1} (x) \,. \end{displaymath} This implies the following universal [[free functor|freeness property]]: \begin{prop} \label{}\hypertarget{}{} Let $\mathfrak{g}$ be an $L_\infty$-algebra. Morphisms of $dg$-algebras $W(\mathfrak{g}) \to A$ are in natural bijection to morphisms of [[graded vector space]]s $\mathfrak{g}^* \to A$. \end{prop} \begin{proof} Forgetting the differential, $W(\mathfrak{g})$ is the free graded-commutative algebra generated by (a shifted copy of) $\mathfrak{g}^*$ and $\mathfrak{g}^*[1]$. Therefore, \begin{displaymath} Hom_{dgca}(W(\mathfrak{g}),A)\subseteq Hom_{gca}(W(\mathfrak{g}),A)=Hom_{grVect}(\mathfrak{g}^*,A)\oplus Hom_{grVect}( \mathfrak{g}^*[1],A). \end{displaymath} Projecting down to $Hom_{grVect}(\mathfrak{g}^*,A)$, one obtains a natural map \begin{displaymath} Hom_{dgca}(W(\mathfrak{g}),A)\to Hom_{grVect}(\mathfrak{g}^*,A), \end{displaymath} which is a bijection. To prove injectivity, we just have to show that the restriction of a dgca morphism $f:W(\mathfrak{g})\to A$ to $\mathfrak{g}^*$ determines the restriction of $f$ to $\mathfrak{g}^*[1]$. One has, for any $x=\sigma(t)\in \mathfrak{g}^*[1]$, \begin{displaymath} \begin{aligned} f(x)&=f(\sigma(t))=f(d_{W(\mathfrak{g})}t-d_{CE(\mathfrak{g})}t)\\ &=d_A f(t)- f(d_{CE(\mathfrak{g})}t). \end{aligned} \end{displaymath} Since $d_{CE(\mathfrak{g})}(t)$ lies in the sub-gca of $W(\mathfrak{g})$ generated by $\mathfrak{g}^*$, the element $f(d_{CE(\mathfrak{g})}(t))$, and therefore $f(x)$, is determined by $f\vert_{\mathfrak{g}^*}$. Next we show surjectivity, i.e. that every morphism of graded vector spaces $\phi:\mathfrak{g}^*\to A$ can be extended to a dgca morphism $f:W(\mathfrak{g})\to A$. Denote by $f_0: \wedge^\bullet \mathfrak{g}^*\to A$ the extension of $\phi$ to a graded commutative algebra morphism, and let $\psi:\mathfrak{g}^*[1]\to A$ be the graded vector space morphism defined by \begin{displaymath} \psi(x)=d_A \phi(t)-f_0d_{CE(\mathfrak{g})}(t), \end{displaymath} for any $x=\sigma(t)\in \mathfrak{g}^*[1]$. The graded vector space morphism $\phi+\psi:\mathfrak{g}^*\oplus\mathfrak{g}^*[1]\to A$ extends to a commutative graded algebra $f:W(\mathfrak{g})\to A$, whose restriction to $\mathfrak{g}^*$ is $\phi$. We want to show that $f$ is actually a dgca morphism. We only need to test commutativity with the differentials on generators $t\in \mathfrak{g}^*$ and $x=\sigma(t)\in \mathfrak{g}^*[1]$. We have \begin{displaymath} d_A f(t)=d_A\phi(t)=\psi(\sigma(t))+f_0d_{CE(\mathfrak{g})}(t)=f(\sigma(t))+ f d_{CE(\mathfrak{g})}(t)=f d_{W(\mathfrak{g})}(t), \end{displaymath} which in particular implies that $d_A f\vert_{\wedge^\bullet \mathfrak{g}^*}=f d_{W(\mathfrak{g})}\vert_{\wedge^\bullet \mathfrak{g}^*}$, and \begin{displaymath} d_A f(x)= d_A \psi(x) = -d_A f_0d_{CE(\mathfrak{g})}(t)=-d_A f (d_{CE(\mathfrak{g})}(t)). \end{displaymath} Since $d_{CE(\mathfrak{g})}(t)\in \wedge^\bullet \mathfrak{g}^*$, we obtain \begin{displaymath} d_A f(x)= -f d_{W(\mathfrak{g})} (d_{CE(\mathfrak{g})}(t))= -f d_{W(\mathfrak{g})}(d_{W(\mathfrak{g})}(t)-x)=f d_{W(\mathfrak{g})}(x). \end{displaymath} \end{proof} \begin{example} \label{}\hypertarget{}{} For $A=CE(\mathfrak{g})$ the [[Chevalley-Eilenberg algebra]] of $\mathfrak{g}$, the inclusion $\mathfrak{g}^*\hookrightarrow CE(\mathfrak{g})$ induces a canonical surjective dgca morphism $W(\mathfrak{g})\to CE(\mathfrak{g})$. This is the identity on the unshifted generators, and 0 on the shifted generators. \end{example} \begin{example} \label{}\hypertarget{}{} For $A = \Omega^\bullet(X)$ the [[de Rham complex]] of a [[smooth manifold]] $X$, we have that \begin{displaymath} Hom_{dgAlg}(W(\mathfrak{g}), \Omega^\bullet(X)) = (\Omega^\bullet(X) \otimes \mathfrak{g})^1 \end{displaymath} is the collection of total degree 1 [[differential form]]s with values in the $\infty$-Lie algebra $\mathfrak{g}$. A morphism of \begin{displaymath} (A, F_A) : W(\mathfrak{g}) \to \Omega^\bullet(X) \end{displaymath} sends the unshifted generators $t^a$ to differential forms $A^a$, which one thinks of as local connection forms, and sends the shifted generators $\sigma t^a$ to their [[curvature]]. The respect for the differential on the shifted generators is the [[Bianchi identity]] on these curvatures. A morphism $W(\mathfrak{g}) \to \Omega^\bullet(X)$ encodes a collection of \emph{flat} $L_\infty$-algebra valued forms precisely if it factors by the canonical morphism $W(\mathfrak{g}) \to CE(\mathfrak{g})$ from above through the [[Chevalley-Eilenberg algebra]] of $\mathfrak{g}$. \end{example} The freeness property of the Weil algebra can be made more explicit by exhibiting a concrete [[isomorphism]] to the free dg-algebra on $\mathfrak{g}^*$. \begin{defn} \label{}\hypertarget{}{} The \emph{canonical free dg-algebra} on $\mathfrak{g}^*$ is \begin{displaymath} F(\mathfrak{g}) := \wedge^\bullet( \mathfrak{g}^* \oplus \mathfrak{g}^*[1], d_F ) \end{displaymath} where the differential $d_f$ is on the unshifted generators $t \in \mathfrak{g}^*$ the shift isomorphism $\sigma : \mathfrak{g}^* \to \mathfrak{g}^*[1]$ extended as a derivation and vanishes on the shifted generators \begin{displaymath} d_F : t \mapsto \sigma(t) \,, \end{displaymath} \begin{displaymath} d_F : \sigma(t) \mapsto 0 \,. \end{displaymath} \end{defn} Or in other words, if $\bar \mathfrak{g}$ is the $\infty$-Lie algebra whose underlying graded vector space is that of $\mathfrak{g}$, but all whose brackets vanish, then \begin{displaymath} F(\mathfrak{g}) = W(\bar \mathfrak{g}) \,. \end{displaymath} Notice the evident \begin{lemma} \label{}\hypertarget{}{} The [[cochain cohomology]] of $F(\mathfrak{g})$ vanishes in positive degree. \end{lemma} To see this, let $K := \sigma^{-1} : F(\mathfrak{g}) \to F(\mathfrak{g})$ be the degree down-shift isomorphism $\mathfrak{g}^*[1] \to \mathfrak{g}^*$ extended as a graded derivation of degree -1, then \begin{displaymath} [d_{F(\mathfrak{g})}, K] = Id : F(\mathfrak{g}) \to F(\mathfrak{g}) \end{displaymath} and hence for any $\omega \in F(\mathfrak{g})$ such that $d_{F(\mathfrak{g})} \omega = 0$ we have $\omega = d_{F(\mathfrak{g})} K \omega$. \begin{prop} \label{}\hypertarget{}{} Given $\mathfrak{g}$, there is an [[isomorphism]] of [[dg-algebra]]s \begin{displaymath} f : F(\mathfrak{g}) \to W(\mathfrak{g}) \end{displaymath} given by \begin{displaymath} f : t \mapsto t \end{displaymath} \begin{displaymath} f : \sigma(t) \mapsto d_{W(\mathfrak{g})} t = d_{CE(\mathfrak{g})} t + \sigma(t) \,. \end{displaymath} \end{prop} \begin{proof} It is clear that $f$ is a dg-algebra homomorphism. The inverse dg-algebra morphism is given on generators by \begin{displaymath} f^{-1} : t \mapsto t \end{displaymath} \begin{displaymath} f^{-1} : \sigma(t) \mapsto \sigma(t) - d_{CE(\mathfrak{g})}(t) \,. \end{displaymath} Note that the isomorphism $f$ is precisely the dgca isomorphism induced between $W(\overline\mathfrak{g})$ and $W(\mathfrak{g})$ by the identity of $\mathfrak{g}^*$ as a graded vector spaces morphism $\overline{\mathfrak{g}}^*\to\mathfrak{g}^*$. \end{proof} \hypertarget{corollary}{}\paragraph*{{Corollary}}\label{corollary} The [[cochain cohomology]] of the Weil algebra of an $L_\infty$-algebra is trivial. \begin{remark} \label{}\hypertarget{}{} This means that [[homotopy theory|homotopy-theoretically]] the Weil algebra is the point. Dually, the $\infty$-Lie algebra $inn(\mathfrak{g})$ is a model for the point. In fact, one can see that $inn(\mathfrak{g})$ is the [[universal principal ∞-bundle]] over $\mathfrak{g}$ in the canonical [[model category|model]] for the [[(∞,1)-topos]] [[SynthDiff∞Grpd]]. In fact, it is a [[groupal model for universal principal ∞-bundles]]. This is discussed at [[∞-Lie algebra cohomology]]. \end{remark} \hypertarget{CharacterizationInSmoothTopos}{}\subsubsection*{{Characterization in the smooth $\infty$-topos}}\label{CharacterizationInSmoothTopos} The Weil algebra of a Lie algebra is naturally identified with the de Rham algebra of differential forms on the ``universal $G$-principal bundle with connection'' in its stacky incarnation (\hyperlink{FreedHopkins13}{Freed-Hopkins 13}): Write $\mathbf{B}G_{conn}\simeq \mathbf{\Omega}(-,\mathfrak{G})///G$ for the universal [[moduli stack]] of $G$-[[principal connections]] (as discussed there), a [[smooth groupoid]]. The quotient projection may be regarded as th universal $G$-connection: \begin{displaymath} \itexarray{ && \mathbf{\Omega}_{flat}(-,\mathfrak{g}) \\ && \downarrow \\ \mathbf{E}G_{conn} &\coloneqq & \mathbf{\Omega}(-,\mathfrak{g}) \\ \downarrow && \downarrow \\ \mathbf{B}G_{conn} &\coloneqq &\mathbf{\Omega}(-,\mathfrak{g})//G } \end{displaymath} (After forgetting the connection/form data this is just the [[universal principal bundle]] $\mathbf{E}G \to \mathbf{B}G$) The differential $k$-forms on a [[smooth groupoid]] $X$ are just homs $X \to \mathbf{\Omega}^k(-)$ into the sheaf of $k$-forms. (See at [[geometry of physics -- differential forms]]). These $\Omega^k(X)$ inherit the de Rham differential and hence form the de Rham complex of the stack. (Notice that this is very different from the hom of $X$ into a shift of the full de Rham complex regarded as a sheaf of complexes. The latter is instead a model for the real [[ordinary cohomology]] of $X$, see at \emph{[[smooth infinity-groupoid -- structures]]} for more on this). One finds (\hyperlink{FreedHopkins13}{Freed-Hopkins 13}) that the de Rham complex, in this sense, of $\mathbf{E}G_{conn}$ is the Weil algebra: \begin{displaymath} \Omega^\bullet(\mathbf{E}G_{conn}) \coloneqq \Omega^\bullet( \mathbf{\Omega}(-,\mathfrak{g}) ) \simeq W(\mathfrak{g}) \,. \end{displaymath} [[!include Weil algebra abstractly -- table]] Turning this around, this motivates to algebraically \emph{define} the [[connection on a principal ∞-bundle]], \href{connection+on+a+smooth+principal+infinity-bundle#ByLieIntegration}{via Lie integration}, as discussed there. \hypertarget{relation_to_cartan_model_for_equivariant_de_rham_cohomology}{}\subsubsection*{{Relation to Cartan model for equivariant de Rham cohomology}}\label{relation_to_cartan_model_for_equivariant_de_rham_cohomology} The Weil algebra may be identified with the [[Cartan model]] for [[equivariant de Rham cohomology]] for the special case of the Lie group $G$ acting on itself by right multiplication. Concersely, the [[Cartan models]] form a generalization of the Weil algebra. See at \emph{\href{equivariant+de+Rham+cohomology#TheCartanModel}{equivariant de Rham cohomology -- Cartan model}} for more. \hypertarget{AsInnerDer}{}\subsection*{{As the CE-algebra of the $L_\infty$-algebra of inner derivations}}\label{AsInnerDer} By the discussion at [[∞-Lie algebra]] and [[Chevalley-Eilenberg algebra]], we may \emph{identify} the [[full subcategory]] of the [[opposite category]] [[dgAlg]] on commutative [[semi-free dga]]s in non-negative degree with that of [[∞-Lie algebra]]s/[[∞-Lie algebroid]]s. That means that the Weil algebra $W(\mathfrak{g})$ of some [[L-∞ algebra]] $\mathfrak{g}$ is the Chevalley-Eilenberg algebra of \emph{another} $\infty$-Lie algebra. \begin{defn} \label{}\hypertarget{}{} For any $\infty$-Lie algebra $\mathfrak{g}$ write $inn(\mathfrak{g})$ for the $\infty$-Lie algebra whose CE-algebra is $W(\mathfrak{g})$: \begin{displaymath} CE(inn(\mathfrak{g})) := W(\mathfrak{g}) \,. \end{displaymath} \end{defn} In the following we discuss these \emph{inner automorphism $\infty$-Lie algebras} in more detail. (See section 6 of (\hyperlink{SSSI}{SSSI})). \hypertarget{for_an_ordinary_lie_algebra}{}\subsubsection*{{For an ordinary Lie algebra}}\label{for_an_ordinary_lie_algebra} \begin{lemma} \label{}\hypertarget{}{} For $\mathfrak{g}$ an ordinary [[Lie algebra]] the [[inner derivation Lie 2-algebra]] is the [[strict Lie 2-algebra]] given by the [[dg-Lie algebra]] \begin{displaymath} inn(\mathfrak{g}) = ( \mathfrak{g} \stackrel{d}{\to} \mathfrak{g}, [-,-]) \end{displaymath} whose \begin{itemize}% \item elements in degree -1 are the elements $x \in \mathfrak{g}$, thought of as inner degree-(-1) [[derivation]]s $\iota_x : CE(\mathfrak{g}) \to CE(\mathfrak{g})$ given by contraction with $x$; \item elements in degree 0 are the derivations of degree 0 that are of the form $\mathcal{L}_X := [d_{CE(\mathfrak{g})}, \iota_x] : CE(\mathfrak{g}) \to CE(\mathfrak{g})$; \item the differential $d = [d_{CE}, -] : \mathfrak{g} \to \mathfrak{g}$ is the commutator of derivations with the differential $d_{CE(\mathfrak{g})}$; \item the bracket is the graded commutator of derivations. \end{itemize} \end{lemma} Equivalently this is identified with the [[differential crossed module]] $(\mathfrak{g} \stackrel{Id}{\to} \mathfrak{g})$ with the action being the [[adjoint action]] of $\mathfrak{g}$ on itself. One checks that for all $x, y \in \mathfrak{g}$ we have in $inn(\mathfrak{g})$ the brackets \begin{itemize}% \item $[\iota_x, \iota_y] = 0$ \item $[\mathcal{L}_x, \iota_y] = \iota_{[x,y]}$ \item $[\mathcal{L}_x, \mathcal{L}_y] = \mathcal{L}_{[x,y]}$ \end{itemize} and of course \begin{itemize}% \item $\mathcal{L}_x = [d, \iota_x]$. \end{itemize} These identities are known as [[Cartan calculus]]. In this context $\mathcal{L}_x$ is called a [[Lie derivative]]. In this sense one may understand $inn(\mathfrak{g})$ for general $\infty$-Lie algebras $\mathfrak{g}$ as providing an $\infty$-version of [[Cartan calculus]]. \hypertarget{relation_to_other_concepts}{}\subsection*{{Relation to other concepts}}\label{relation_to_other_concepts} \hypertarget{LieAlgValuedForms}{}\subsubsection*{{$\infty$-Lie algebra valued differential forms}}\label{LieAlgValuedForms} For $\mathfrak{g}$ an [[∞-Lie algebra]], $X$ a [[smooth manifold]], an [[∞-Lie algebra valued differential form]] is a morphism \begin{displaymath} \Omega^\bullet(X) \leftarrow W(\mathfrak{g}) : A \end{displaymath} of [[dg-algebra]]s, from the Weil algebra into the [[de Rham complex]] of $X$. The image of the unshifted generators $A : \wedge^1 \mathfrak{g}^* \to \Omega^\bullet(X)$ are the forms themselves, the image of the shifted generators $F_A : \wedge^1 \mathfrak{g}^*[1]$ are the corresponding [[curvature]]s. The respect for the differential on the shifted generators are the [[Bianchi identity]] on the curvatures. Precisely if the curvatures vanish does the morphism factor through the [[Chevalley-Eilenberg algebra]] $W(\mathfrak{g}) \to CE(\mathfrak{g})$. \begin{displaymath} (F_A = 0) \;\;\Leftrightarrow \;\; \left( \itexarray{ && CE(\mathfrak{g}) \\ & {}^{\mathllap{\exists A_{flat}}}\swarrow & \uparrow \\ \Omega^\bullet(X) &\stackrel{A}{\leftarrow}& W(\mathfrak{g}) } \right) \,. \end{displaymath} \hypertarget{invariant_polynomials_and_chernsimons_elements}{}\subsubsection*{{Invariant polynomials and Chern-Simons elements}}\label{invariant_polynomials_and_chernsimons_elements} A [[cocycle]] in the [[∞-Lie algebra cohomology]] of the [[∞-Lie algebra]] $\mathfrak{g}$ is a closed element in the [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{g})$. An [[invariant polynomial]] $\langle -\rangle$ on $\mathfrak{g}$ is a closed element in the Weil algebra $\langle -\rangle \in W(\mathfrak{g})$, subject to the additional condition that it its entirely in the shifted copy of $\mathfrak{g}$, $\langle - \rangle \in \wedge^\bullet (\mathfrak{g}^*[1])$. \begin{displaymath} \langle -\rangle \in \wedge^\bullet( \mathfrak{g}^*[1] ) \end{displaymath} \begin{displaymath} d_{W(\mathfrak{g})} \langle -\rangle = 0 \,. \end{displaymath} For $x \in \mathfrak{g}$ an element of the $\infty$-Lie algebra, let \begin{displaymath} \iota_x : W(\mathfrak{g}) \to W(\mathfrak{g}) \end{displaymath} the evident operation of contraction with $x$ \begin{displaymath} \iota_x : t \mapsto t(x) \end{displaymath} \begin{displaymath} \iota_x : \sigma(t) \mapsto 0 \end{displaymath} extended as a graded derivation. Then the [[Lie derivative]] \begin{displaymath} \mathcal{L}_x := ad_x := [d_{W(\mathfrak{g})}, \iota_x] : W(\mathfrak{g}) \to W(\mathfrak{g}) \end{displaymath} encodes the coadjoint action of $\mathfrak{g}$ on $\mathfrak{g}^*$. By the above definition of an [[invariant polynomial]] $\langle - \rangle$, we have \begin{displaymath} \iota_x \langle - \rangle = 0 \end{displaymath} and \begin{displaymath} d_{W(\mathfrak{g})} \langle - \rangle = 0 \end{displaymath} and hence \begin{displaymath} ad_x \langle -\rangle = 0 \,. \end{displaymath} Since the cohomology of $W(\mathfrak{g})$ is trivial, there is necessarily for each invariant polynomial an element $cs_{\langle -\rangle}$ such that \begin{displaymath} d_{W(\mathfrak{g})} cs_{\langle -\rangle} = \langle -\rangle \,. \end{displaymath} This is the [[Chern-Simons element]] of the invariant polynomial. Notice, crucially, that this is ingeneral \emph{not} restricted to the shifted part $\wedge^\bullet (\mathfrak{g}^*[1])$ Its restriction \begin{displaymath} \mu_{\langle -\rangle} := cs_{\langle - \rangle}|_{\wedge^\bullet \mathfrak{g}^*} \end{displaymath} to the unshifted copy, hence to the [[Chevalley-Eilenberg algebra]], is the cocycle that is in transgression with $\langle - \rangle$. For \begin{displaymath} (A,F_A) : W(\mathfrak{g}) \to \Omega^\bullet(X) \end{displaymath} a collection of $\mathfrak{g}$-valued differential forms (as \href{LieAlgValuedForms}{above}) and $\langle -\rangle : CE(b^{n-1}\mathbb{R}) \to W(\mathfrak{g})$ an [[invariant polynomial]], the composite \begin{displaymath} \langle F_A\rangle : CE(b^{n-1}\mathbb{R}) \stackrel{\langle - \rangle}{\to} W(\mathfrak{g}) \stackrel{(A,F_A)}{\to} \Omega^\bullet(X) \end{displaymath} is the corresponding [[curvature characteristic form]], a closed $n$-form on $X$. For $(\langle - \rangle, cs) : W(b^{n-1}) \to W(\mathfrak{g})$ the corresponding [[Chern-Simons element]] we have that $cs(A,F_A)$ is the corresponding [[Chern-Simons form]] on $X$. \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \hypertarget{WeilofLieAlg}{}\subsubsection*{{Weil algebra of a Lie algebra}}\label{WeilofLieAlg} Let $\mathfrak{g}$ be a finite dimensional [[Lie algebra]]. This Lie algebra regarded as a [[Lie algebroid]] has as base manifold the point, $X_0 = pt$. Its algebra of functions is accordingly the ground field, and the algebra $\wedge^\bullet_{C^\infty(X_0)} \mathfrak{g}^*$ is just a [[Grassmann algebra]]. The [[Chevalley-Eilenberg algebra]] is \begin{displaymath} CE(\mathfrak{g}) = (\wedge^\bullet \mathfrak{g}^*, d_{\mathfrak{g}}) \,, \end{displaymath} where the differential acts on the elements of $\mathfrak{g}^*$ in degree 1 by the linear dual of the Lie bracket. \begin{displaymath} d \mathfrak{g}|_{\mathfrak{g}^*} = [-,-]^* : \mathfrak{g}^* \to \mathfrak{g}^* \wedge \mathfrak{g}^* \,. \end{displaymath} The corresponding Weil algebra is obtained by adding another copy of $\mathfrak{g}^*$ in degree 2 \begin{displaymath} W(\mathfrak{g}) = (\wedge^\bullet (\mathfrak{g}^* \oplus \mathfrak{g}^*[1]), d_{W(\mathfrak{g})}) \end{displaymath} where with $\sigma : \mathfrak{g}^* \to \mathfrak{g}^*[1]$ the degree shift isomorphism, the differential acts as \begin{displaymath} d_{W(\mathfrak{g})}|_{\mathfrak{g}^*} : [-,-]^* + \sigma \end{displaymath} \begin{displaymath} d_{W(\mathfrak{g})}|_{\mathfrak{g}^*[1]} : \sigma \circ d_{CE(\mathfrak{g})} \circ \sigma^{-1} \,. \end{displaymath} For illustration, we spell this out in a basis. Let $\{t_a\}_a$ be a basis for the underlying vector space of $\mathfrak{g}$ and let $\{C^a{}_{b c}\}$ be the corresponding structure constants of the Lie bracket \begin{displaymath} [t_b, t_c] = C^a{}_{b c} t_a \,. \end{displaymath} Then the Chevalley-Eilenberg algebra is generated on generators $\{t^a\}$ of degree 1, on which the differential acts as \begin{displaymath} d_{CE(\mathfrak{g})} : t^a \mapsto - \frac{1}{2} C^a{}_{b c} t^b \wedge t^c \,. \end{displaymath} The Weil algebra in turn is generated from these generators $\{t^a\}$ in degree 1 and generators $\{r^a\}$ in degree 2, with differential given by \begin{displaymath} d_{W(\mathfrak{g})} : t^a \mapsto - \frac{1}{2} C^a{}_{b c} t^b \wedge t^c + r^a \end{displaymath} \begin{displaymath} d_{W(\mathfrak{g})} : r^a \mapsto C^a{}_{b c} t^b \wedge r^c \,. \end{displaymath} \hypertarget{weil_algebra_of_a_0lie_algebroid}{}\subsubsection*{{Weil algebra of a 0-Lie algebroid}}\label{weil_algebra_of_a_0lie_algebroid} A 0-[[truncated]] Lie algebroid is one for which the chain complex of modules over the $T$-algebra in degree 0 vanishes: \begin{displaymath} CE(\mathfrak{a}) = (\wedge^\bullet_A (\mathfrak{a}^*), d_{CE(\mathfrak{a})}) = (A, d = 0) \,. \end{displaymath} For instance for $T$=[[CartSp]] the theory of [[smooth algebra]]s, any [[smooth manifold]] $X$ regarded as an [[L-∞ algebroid]] is a 0-Lie algebroid with $CE(X) = C^\infty(X)$ the [[smooth algebra]] of [[smooth function]]s on $X$. \begin{prop} \label{}\hypertarget{}{} The Weil algebra of a 0-Lie algebroid $X$ is the [[Kähler differential|Kähler]] [[de Rham complex]] of $A = CE(X)$: \begin{displaymath} W(\mathfrak{a}) = (\Omega^\bullet(A), d_{dR}) \,. \end{displaymath} \end{prop} This Weil algebra is the Chevalley-Eilenberg algebra of the [[tangent Lie algebroid]] $T X$ of $X$, which is the [[de Rham algebra]] $\Omega^\bullet(X)$ of $X$: \begin{displaymath} W(X) = CE(T X) = (\Omega^\bullet(X), d_{dR}) \,. \end{displaymath} \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[∞-Lie algebra cohomology]] \item [[Chevalley-Eilenberg algebra]] \item \textbf{Weil algebra} \item [[invariant polynomial]] \item [[Sullivan model of free loop space]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} Among the original references on Weil algebras for ordinary Lie algebras is \begin{itemize}% \item [[Henri Cartan]], \emph{Cohomologie r\'e{}elle d'un espace fibr\'e{} principal diffrentielle} I, II, S\'e{}minaire Henri Cartan, 1949/50, pp. 19-01 -- 19-10 and 20-01 -- 20-11, CBRM, (1950). \end{itemize} and \begin{itemize}% \item [[Henri Cartan]], \emph{Notions d'alg\'e{}bre diff\'e{}rentielle; application aux groupes de Lie et aux vari\'e{}t\'e{}s ou op\`e{}re un groupe de Lie} , Colloque de topologie (espaces fibrs), Bruxelles, (1950), pp. 15--27. \end{itemize} This also explains the use of the Weil algebra in the calculation of the [[equivariant cohomology|equivariant]] [[de Rham cohomology]] of manifolds acted on by a compact group. These papers are reprinted, explained and put in a modern context in the book A clasical textbook account of standard material is in chapter VI, vol III of \begin{itemize}% \item [[Werner Greub]], [[Stephen Halperin]], [[Ray Vanstone]], \emph{[[Connections, Curvature, and Cohomology]]} Academic Press (1973) \end{itemize} Some remarks on the notation there as compared to ours: our $d_W$ is their $\delta_W$ on p. 226 (vol III). Their $\delta_E$ is our $d_{CE}$. Their $\delta_\theta$ is our $d_\rho$ ($\theta$/$\rho$ denoting the representation).. In the context of [[equivariant de Rham cohomology]]: \begin{itemize}% \item [[Michael Atiyah]], [[Raoul Bott]], \emph{The moment map and equivariant cohomology}, Topology 23, 1 (1984) (, \href{https://www.math.stonybrook.edu/~mmovshev/MAT570Spring2008/BOOKS/atiyahbott_moment.pdf}{pdf}) \item Jaap Kalkman, \emph{BRST model applied to symplectic geometry}, Ph.D. Thesis, Utrecht, 1993 (\href{https://arxiv.org/abs/hep-th/9308132}{arXiv:hep-th/9308132}, \href{http://cds.cern.ch/record/568522}{cds:9308132}, \href{http://projecteuclid.org/euclid.cmp/1104252784}{euclid:1104252784}) \end{itemize} and with an eye towards [[supersymmetry]]: \begin{itemize}% \item Mauri Miettinen, \emph{Weil Algebras and Supersymmetry} (\href{https://arxiv.org/abs/hep-th/9612209}{arXiv:hep-th/9612209}, \href{http://cds.cern.ch/record/317377}{cds:317377}, \href{http://inspirehep.net/record/427720}{spire:427720}) \item [[Victor Guillemin]], [[Shlomo Sternberg]], \emph{Supersymmetry and equivariant de Rham theory}, Springer, (1999) (\href{https://link.springer.com/book/10.1007/978-3-662-03992-2}{doi:10.1007/978-3-662-03992-2}) \end{itemize} The (obvious but conceptually important) observation that [[Lie algebra-valued 1-forms]] regarded as morphisms of graded vector spaces $\Omega^\bullet(X) \leftarrow \wedge^1 \mathfrak{g}^* : A$ are equivalently morphisms of dg-algebras out of the Weil algebra $\Omega^\bullet(X) \leftarrow W(\mathfrak{g}) : A$ and that one may think of as the identity $W(\mathfrak{g}) \leftarrow W(\mathfrak{g}) : Id$ as the \emph{universal $\mathfrak{g}$-connection} appears in early articles for instance highlighted on p. 15 of \begin{itemize}% \item Franz W. Kamber; Philippe Tondeur, \emph{Semisimplicial Weil algebras and characteristic classes for foliated bundles in Cech cohomology} , Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Stanford Univ., Stanford, Calif., 1973), Part 1, pp. 283--294. Amer. Math. Soc., Providence, R.I., (1975). \end{itemize} A survey of Weil algebras for Lie algebras is also available at \begin{itemize}% \item [[eom|Encyclopedia of Mathematics]]: \href{http://eom.springer.de/W/w130050.htm}{Weil algebra of a Lie algebra} \end{itemize} Weil algebra for [[L-infinity algebra]]s and their role in defining [[invariant polynomial]]s and [[Chern-Simons element]]s on $\infty$-Lie algebras from [[infinity-Lie algebra cohomology|L-infinity algebra cocycle]] are considered in \begin{itemize}% \item [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]], \emph{$L_{\infty}$ algebra connections and applications to String- and Chern-Simons $n$-transport} () \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} Further discussion of Weil algebras for the [[string Lie 2-algebra]]: \begin{itemize}% \item [[Lennart Schmidt]], \emph{Twisted Weil Algebras for the String Lie 2-Algebra}, in [[Christian Saemann]], [[Urs Schreiber]], [[Martin Wolf]] (eds.) \emph{\href{http://www.maths.dur.ac.uk/lms/109/index.html}{Higher Structures in M-Theory}}, \href{http://www.maths.dur.ac.uk/lms/}{Durham Symposium} 2018, Fortschritte der Physik 2019 (\href{https://arxiv.org/abs/1903.02873}{arXiv:1903.02873}) \end{itemize} [[!redirects Weil algebras]] \end{document}