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\begin{document}

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


\section*{curved L-infinity 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]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \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}

A \emph{curved $L_\infty$-algebra} (e.g. \hyperlink{Markl11}{Markl 11, p. 100}) is just like an ordinary \emph{[[L-∞ algebra]]}, but possibly including also a 0-ary bracket, i.e. a constant. Conversely an ordinary [[L-∞ algebra]] is a curved $L_\infty$-algebra for which the 0-ary operation happens to be zero.

Accordingly, a ``strong homotopy [[homomorphism]]'' of curved $L_\infty$-algebras is defined just as for ordinary $L_\infty$-algebras, but allowing also for a 0-ary component. Notice that such ``curved sh-maps'' may be non-trivial even between ordinary $L_\infty$-algebras (amplified e.g. in \hyperlink{MehtaZambon12}{Mehta-Zambon 12, below (2)}).

The dual [[Chevalley-Eilenberg algebras]] automatically capture curved $L_\infty$-algebras unless one imposes a constraint: the non-curved $L_\infty$-algebras correspond to the [[augmented algebra|augmented]] CE-algebras. Similarly in the [[dg-coalgebra]] description the restriction to non-curved $L_\infty$-algebras requires co-augmentation or else (this is what is commonly used) non-unital dg-coalgebras.

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

Where an ordinary \emph{[[L-infinity algebra]]} is a $\mathbb{Z}$-[[graded vector space]] $\mathfrak{g}$ equipped for all $n \in \mathbb{N}$, $n \geq 1$ with $n$-ary brackets:

\begin{displaymath}
l_n \;\colon\; \mathfrak{g}^{\otimes^n} \longrightarrow \mathfrak{g}
\end{displaymath}
out of the [[tensor product]] of $n$-copies of $\mathfrak{g}$, subject to some conditions, for a curved $L_\infty$-algebra also a component

\begin{displaymath}
l_0 \;\colon\; \mathfrak{g}^{\otimes^0} \simeq \mathbb{R} \to \mathfrak{g}
\end{displaymath}
is allowed allowed. Since an $\mathbb{R}$-linear map out of $\mathbb{R}$ is uniquely fixed by a single element (the image of $1 \in \mathbb{R}$), this is ``a constant'', called the \emph{curvature} of the curved $L_\infty$-algebra.

Now the strong homotopy Jacobi identity

\begin{equation}
\sum_{{i,j \in \mathbb{N}} \atop {i+j = n+1}} 
  \sum_{\sigma \in UnShuff(i,j)}
  \chi(\sigma,v_1, \cdots, v_{n})
  (-1)^{i(j-1)}
   l_{j} \left(
     l_i \left( v_{\sigma(1)}, \cdots, v_{\sigma(i)} \right),
     v_{\sigma(i+1)} , \cdots , v_{\sigma(n)}
   \right)
  = 0
  \,,
\label{LInfinityJacobiIdentity}\end{equation}
implies in particular that

\begin{displaymath}
l_1 \circ l_1 = \pm l_2(l_0, -)
\end{displaymath}
hence that the unary operation $l_1$ no longer necessarily squares to zero (no longer defines a [[chain complex]] $(\mathfrak{g}, l_1)$) but to the binary bracket with the curvature.

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

\begin{itemize}%
\item [[curved A-infinity algebra]]


\item [[curved dg-algebra]]



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

\begin{itemize}%
\item [[Martin Markl]], \emph{Deformation Theory of Algebras and Their Diagrams}, Regional Conference Series in Mathematics Number 116, American Mathematical Society (2011)


\item [[Andrey Lazarev]], Travis Schedler, \emph{Curved infinity-algebras and their characteristic classes}, J Topology (2012) 5 (3): 503-528 (\href{https://arxiv.org/abs/1009.6203}{arXiv:1009.6203})


\item [[Rajan Mehta]], [[Marco Zambon]], \emph{$L_\infty$-Actions}, Differential Geometry and its Applications 30 (2012), 576-587 (\href{https://arxiv.org/abs/1202.2607}{arXiv:1202.2607})



\end{itemize}
[[!redirects curved L-infinity algebras]]

[[!redirects curved L-∞ algebra]] [[!redirects curved L-∞ algebras]]

[[!redirects curved sh-map]] [[!redirects curved sh-maps]]

[[!redirects curved L-infinity homomorphism]] [[!redirects curved L-infinity homomorphisms]]

[[!redirects curved L-∞ homomorphism]] [[!redirects curved L-∞ homomorphisms]]



\end{document}