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\begin{document}

%-------------------------------------------------------------------


\section*{automorphism infinity-Lie algebra}

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

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

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

\hypertarget{rational_homotopy_theory}{}\paragraph*{{Rational homotopy theory}}\label{rational_homotopy_theory}

[[!include differential graded objects - contents]]

\begin{quote}%
\ldots{} under construction \ldots{}


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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{AutomorphismGroup}{Automorphism group}\dotfill \pageref*{AutomorphismGroup} \linebreak
\noindent\hyperlink{ClassifyingSpaces}{Classifying space for $Aut(X)$-principal bundles}\dotfill \pageref*{ClassifyingSpaces} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \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{automorphism $\infty$-Lie algebra} $aut(\mathfrak{g})$ of an [[∞-Lie algebra]] $\mathfrak{g}$ -- or dually $aut(CE(\mathfrak{g}))$ of the corresponding [[Chevalley-Eilenberg algebra]] -- has in degree $k$ the [[derivations]] on $CE(\mathfrak{g})$ of degree $-k$. The [[higher Lie algebra]] version of the [[automorphism Lie algebra]] of an ordinary [[Lie algebra]].

In terms of [[rational homotopy theory]] $aut(\mathfrak{g})$ is a model for the rationalization of the group of [[automorphisms]]s of the [[rational space]] $\exp(\mathfrak{g})$ corresponding to $CE(\mathfrak{g})$ under the [[Sullivan construction]].

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

Let $A := (\wedge^\bullet \mathfrak{a}^*, d_A)$ be a [[semifree dga|semifree]] [[dg-algebra]] of [[finite type]].

Notice that for $\phi : A \to A$ a [[derivation]] of degree $-k$ and $\lambda : A\to A$ another derivation of degree $-l$ the [[commutator]]

\begin{displaymath}
[\phi,\lambda] := \phi \circ \lambda - \lambda \circ \phi : A \to A
\end{displaymath}
is itself a derivation, of degree $-(k+l)$. In particular, since the [[differential]] $d_A : A \to A$ is itself a derivation of degree +1, we have that

\begin{displaymath}
d_A \phi := [d_A, \phi] : A \to A
\end{displaymath}
is a derivation of degree $-(k+1)$.

\begin{udefn}
\textbf{(automorphism $\infty$-Lie algebra)}

The [[∞-Lie algebra]] $aut(A)$ is the [[dg-Lie algebra]] which

\begin{itemize}%
\item in degree $-k$ for $k \gt 0$ has the derivations $\phi : A \to A$ of degree $-k$;


\item in degree $0$ the derivations that commute with the differential $d_A$


\item whose differential $\delta_{aut(A)} := [d_A,-]$ is given by the commutator with the differential of $A$;


\item whose Lie bracket is the commutator $[\phi,\lambda] = \phi \circ \lambda - \lambda \circ \phi$.



\end{itemize}
\end{udefn}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{AutomorphismGroup}{}\subsubsection*{{Automorphism group}}\label{AutomorphismGroup}

For stating the fundamental theorem about $aut(\mathfrak{g})$ below we need some facts about the ordinary [[automorphism group]] of a dg-algebra $A$.

(\ldots{})

(See chapter 6 of \hyperlink{Sullivan}{Sullivan}).

\hypertarget{ClassifyingSpaces}{}\subsubsection*{{Classifying space for $Aut(X)$-principal bundles}}\label{ClassifyingSpaces}

Let $X$ be a [[rational space]] whose [[Sullivan model]] is $\mathfrak{g}$, $X \simeq \exp(\mathfrak{g})$. Let $aut'(\mathfrak{g}) \subset aut(\mathfrak{g})$ be the sub dg-algebra of the automorphism $\infty$-Lie algebra on the maximal nilpotent ideal in degree 0. Let $G(X)$ be the maximal reductive group of genuine automorphisms of $CE(\mathfrak{g})$ (see \hyperlink{AutomorphismGroup}{above}).

Then the [[rational space]]

\begin{displaymath}
\exp(aut'(\mathfrak{g}))/G(X)
  \simeq
  B Aut (X)
\end{displaymath}
is the [[classifying space]] for $Aut(X)$-[[principal bundle]]s, i.e. for [[bundle]]s with typical fiber $X$.

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

\begin{itemize}%
\item The [[inner derivation Lie 2-algebra]] $inn(\mathfrak{g})$ is the full subalgebra of the automorphism $\infty$-Lie algebra on the \emph{inner} derivations of the [[Chevalley-Eilenberg algebra]] of a [[Lie algebra]] $\mathfrak{g}$.

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

\begin{itemize}%
\item [[automorphism Lie algebra]]


\item [[deformation theory]]


\item [[tangent complex]]


\item [[cotangent complex]]



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

The general definition of $aut(\mathfrak{g})$ is the topic of p. 313 (45 of 63) and following in

\begin{itemize}%
\item [[Dennis Sullivan]], \emph{Infinitesimal computations in topology} Publications Math\'e{}matiques de l'IH\'E{}S, 47 (1977) (\href{http://www.numdam.org/item?id=PMIHES_1977__47__269_0}{numdam})

\end{itemize}
The automorphism group $Aut(A)$ of a dg-algebra is discussed in paragraph 6 there. Few details on proofs are given there. Only recently in

\begin{itemize}%
\item [[Andrey Lazarev]], [[Jonathan Block]], \ldots{}

\end{itemize}
a detailed proof is given.

Concrete computations of $aut(\mathfrak{g})$ for some classes of [[rational space]]s $X = \exp(\mathfrak{g})$ can be found for instance in

\begin{itemize}%
\item Samual Bruce Smith, \emph{The rational homotopy Lie algebra of classifying spaces for formal two-stage spaces} , Journal of Pure and Applied Algebra Volume 160, Issues 2-3, 25 June 2001, Pages 333-343

\end{itemize}
[[!redirects automorphism ∞-Lie algebra]] [[!redirects automorphism dg-Lie algebra]]

[[!redirects automorphism infinity-Lie algebras]] [[!redirects automorphism ∞-Lie algebras]] [[!redirects automorphism dg-Lie algebras]]

[[!redirects automorphism L-infinity algebra]] [[!redirects automorphism L-∞ algebra]] [[!redirects automorphism L-infinity algebras]] [[!redirects automorphism L-∞ algebras]]



\end{document}