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\section*{super L-infinity algebra}

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

\hypertarget{supergeometry}{}\paragraph*{{Supergeometry}}\label{supergeometry}

[[!include supergeometry - contents]]

\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{Definition}{Definition}\dotfill \pageref*{Definition} \linebreak
\noindent\hyperlink{history}{History}\dotfill \pageref*{history} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \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}

A \emph{super $L_\infty$-algebra} is an [[L-∞ algebra]] in the context of [[superalgebra]]: the [[higher category theory|higher category theoretical]]/[[homotopy theory|homotopy theoretical]] version of a [[super Lie algebra]]. For more background see at \emph{[[geometry of physics -- superalgebra]]}.

In the [[supergravity]] literature the [[formal duality|formal dual]] of super $L_\infty$-algebras of [[finite type]] came to be known as ``[[FDA]]''s (see remark \ref{SuperLInfintiyAsFDA} below), a decade before plain [[L-∞ algebras]] were discussed in the mathematical literature. The key example in this context are [[extended super Minkowski spacetimes]], which are [[super L-∞ algebras]] obtained by iterated [[universal higher central extension]] from the [[super Minkowski spacetime]] [[super Lie algebra]]. The super-[[L-∞ algebra cohomology]] of these (called ``[[tau-cohomology]]'' in the physics literature) turns out to classify [[super p-branes]] and serves as a tool for the construction of [[supergravity]] theories in the [[D'Auria-Fré formulation of supergravity]]. For more background on this see at \emph{[[geometry of physics -- fundamental super p-branes]]}.

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

Abstractly, the definition is immediate:

\begin{defn}
\label{SuperLInfinityAlgebra}\hypertarget{SuperLInfinityAlgebra}{}
A \textbf{super $L_\infty$-algebra} is an [[L-∞ algebra]] [[internalization|internal to]] the [[symmetric monoidal category]] of [[super vector spaces]] (i.e. in [[chain complexes of super vector spaces]]).

\end{defn}
Explicitly this equivalently comes down to the following definition in components:

\begin{defn}
\label{SuperGradedSignatureOfPermutation}\hypertarget{SuperGradedSignatureOfPermutation}{}
\textbf{(super graded signature of a permutation)}

Let $V$ be a $\mathbb{Z}$-[[graded object|graded]] [[super vector space]], hence a $\mathbb{Z} \times (\mathbb{Z}/2)$-bigraded vector space.

For $n \in \mathbb{N}$ let

\begin{displaymath}
\mathbf{v} = (v_1, v_2, \cdots, v_n)
\end{displaymath}
be an [[n-tuple]] of elements of $V$ of homogeneous degree $(n_i, s_i) \in \mathbb{Z} \times \mathbb{Z}/2$, i.e. such that $v_i \in V_{(n_i,s_i)}$.

For $\sigma$ a [[permutation]] of $n$ elements, write $(-1)^{\vert \sigma \vert}$ for the [[signature of a permutation|signature of the permutation]], which is by definition equal to $(-1)^k$ if $\sigma$ is the composite of $k \in \mathbb{N}$ permutations that each exchange precisely one pair of neighboring elements.

We say that the \emph{super $\mathbf{v}$-graded signature of $\sigma$}

\begin{displaymath}
\chi(\sigma, v_1, \cdots, v_n) \;\in\; \{-1,+1\}
\end{displaymath}
is the product of the [[signature of a permutation|signature of the permutation]] $(-1)^{\vert \sigma \vert}$ with a factor of

\begin{displaymath}
(-1)^{n_i n_j}(-1)^{s_i s_j}
\end{displaymath}
for each interchange of neighbours $(\cdots v_i,v_j, \cdots )$ to $(\cdots v_j,v_i, \cdots )$ involved in the decomposition of the permuation as a sequence of swapping neighbour pairs (see at \emph{[[signs in supergeometry]]} for discussion of this combination of super-grading and homological grading).

\end{defn}
Now def. \ref{SuperLInfinityAlgebra} is equivalent to the following def. \ref{sLInfinityDefinitionViaGeneralizedJacobiIdentity}. This is just the definiton for \hyperlink{L-infinity-algebra#DefinitionViaHigherBrackets}{L-infinity algebras}, with the pertinent sign $\chi$ now given by def. \ref{SuperGradedSignatureOfPermutation}.

\begin{defn}
\label{sLInfinityDefinitionViaGeneralizedJacobiIdentity}\hypertarget{sLInfinityDefinitionViaGeneralizedJacobiIdentity}{}
An \emph{super $L_\infty$-algebra} is

\begin{enumerate}%
\item a $\mathbb{Z} \times (\mathbb{Z}/2)$-[[graded object|graded]] [[vector space]] $\mathfrak{g}$;


\item for each $n \in \mathbb{N}$ a [[multilinear map]], called the \emph{$n$-ary bracket}, of the form

\begin{displaymath}
l_n(\cdots) 
    \;\coloneqq\; 
  [-,-, \cdots, -]_n 
  \;\colon\; 
    \underset{n \; \text{copies}}{\underbrace{\mathfrak{g} \otimes \cdots \otimes \mathfrak{g}}}
    \longrightarrow
    \mathfrak{g}
\end{displaymath}
and of degree $n-2$



\end{enumerate}
such that the following conditions hold:

\begin{enumerate}%
\item (\textbf{super graded skew symmetry}) each $l_n$ is graded antisymmetric, in that for every [[permutation]] $\sigma$ of $n$ elements and for every [[n-tuple]] $(v_1, \cdots, v_n)$ of homogeneously graded elements $v_i \in \mathfrak{g}_{\vert v_i \vert}$ then

\begin{displaymath}
l_n(v_{\sigma(1)}, v_{\sigma(2)},\cdots ,v_{\sigma(n)}) 
  = 
  \chi(\sigma,v_1,\cdots, v_n) \cdot l_n(v_1, v_2, \cdots v_n)
\end{displaymath}
where $\chi(\sigma,v_1,\cdots, v_n)$ is the super $(v_1,\cdots,v_n)$-graded signature of the permuation $\sigma$, according to def. \ref{SuperGradedSignatureOfPermutation};


\item (\textbf{strong homotopy [[Jacobi identity]]}) for all $n \in \mathbb{N}$, and for all [[n-tuple|(n+1)-tuples]] $(v_1, \cdots, v_{n+1})$ of homogeneously graded elements $v_i \in \mathfrak{g}_{\vert v_i \vert}$ the followig [[equation]] holds

\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}
where the inner sum runs over all $(i,j)$-[[unshuffles]] $\sigma$ and where $\chi$ is the super graded signature sign from def. \ref{SuperGradedSignatureOfPermutation}.



\end{enumerate}
A \emph{strict [[homomorphism]]} of super $L_\infty$-algebras

\begin{displaymath}
is
  \mathfrak{g}_1 \longrightarrow \mathfrak{g}_2
\end{displaymath}
is a [[linear map]] that preserves the bidegree and all the brackets, in an evident sens.

A \emph{strong homotopy homomorphism} (``sh map'') of super $L_\infty$-algebras is something weaker than that, best defined in [[formal duality|formal duals]], below in def. \ref{SuperLInfinityCEAlgebra}.

\end{defn}
In order to define the correct homomorphisms between super $L_\infty$-algebras (``sh-maps'') as well as their super-[[L-∞ algebra cohomology]], consider the following dualization of the above definition:

\begin{defn}
\label{SuperLInfinityCEAlgebra}\hypertarget{SuperLInfinityCEAlgebra}{}
A super $L_\infty$ algebra $\mathfrak{g}$ is of \emph{[[finite type]]} if the underlying $\mathbb{Z} \times (\mathbb{Z}/2)$-[[graded vector space]] is degreewise of [[finite number|finite]] [[dimension]].

If $\mathfrak{g}$ is of finite type, then its [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{g})$ is the [[differential graded-commutative superalgebra]] whose underlying [[graded algebra]] is the super-Grassmann algebra

\begin{displaymath}
\wedge^\bullet \mathfrak{g}^{\ast}
\end{displaymath}
of the graded degreewise [[dual vector space]] $\mathfrak{g}^\ast$, equipped with the [[differential]] which on generators is the sum of the [[dual linear maps]] of the $n$-ary brackets:

\begin{displaymath}
d_{\mathfrak{g}}
   \coloneqq
  [-]^\ast + [-,-]^\ast + [-,-,-]^\ast + \cdots
  \;\colon\;
  \wedge^1 \mathfrak{g}^\ast
    \longrightarrow
  \wedge^\bullet \mathfrak{g}^\ast
\end{displaymath}
and extended to all of $\wedge^\bullet \mathfrak{g}^\ast$ as a super-graded [[derivation]] of degree $(1,even)$.

Notice that here the [[signs in supergeometry]] are such that for $\alpha_i \in \mathfrak{g}^\ast_{(n_i,s_i)}$ elements of homogenous bidegree, then

\begin{displaymath}
\alpha_1 \wedge \alpha_2
  \;=\;
  -(-1)^{n_1 n_2} (-)^{s_1 s_2}
\end{displaymath}
and

\begin{displaymath}
d_{\mathfrak{g}}
  (\alpha_1 \wedge \alpha_2)
  \;=\;
  (d_{\mathfrak{g}} \alpha_1) \wedge \alpha_2
  + 
  (-1)^{n_1} \alpha_1 \wedge (d_{\mathfrak{g}} \alpha_2)
  \,.
\end{displaymath}
(see at \emph{[[signs in supergeometry]]} for more on this).

A \emph{strong homotopy homomorphism} (``sh-map'') between super $L_\infty$-algbras of [[finite type]]

\begin{displaymath}
f \;\colon\; \mathfrak{g}_1 \longrightarrow \mathfrak{g}_2
\end{displaymath}
is defined to be a homomorphism of [[dg-algebras]] between their [[Chevalley-Eilenberg algebras]] going the other way:

\begin{displaymath}
CE(\mathfrak{g}_1) \longleftarrow CE(\mathfrak{g}_2)
    \;\colon\;
  f^\ast
\end{displaymath}
(here $f^\ast$ is the primitive concept, and $f$ is defined as the [[formal duality|formal dual]] of $f$). Hence the [[category]] of super $L_\infty$-algebras of [[finite type]] is the [[full subcategory]]

\begin{displaymath}
s L_\infty Alg \hookrightarrow dgcsAlg^{op}
\end{displaymath}
of the [[opposite category]] of [[differential graded-commutative superalgebras]] on those that are CE-algebras as above.

Finally, the [[cochain cohomology]] of the [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{g})$ of a super $L_\infty$ algebra of [[finite type]] is its \emph{[[L-∞ algebra cohomology]]} with [[coefficients]] in $\mathbb{R}$:

\begin{displaymath}
H^\bullet(\mathfrak{g}, \mathbb{R})
   \;=\;
  H^\bullet(CE(\mathfrak{g}))
  \,.
\end{displaymath}
\end{defn}
\begin{remark}
\label{LInfinityTerminology}\hypertarget{LInfinityTerminology}{}
Special cases of the general concept of [[super L-∞ algebras]] def. \ref{sLInfinityDefinitionViaGeneralizedJacobiIdentity} go by special names:

Let $\mathfrak{g}$ be a [[super L-∞ algebra]].

If $\mathfrak{g}$ is concentrated in even $\mathbb{Z}/2$-degree, it is called an \emph{[[L-∞ algebra]]}.

If the only possibly non-vanishing brackets of $\mathfrak{g}$ are the unary one $[-]$ (which induces the structure of a [[chain complex]] on $\mathfrak{g}$) and the binary one, then $\mathfrak{g}$ is equivalently a (super-)[[dg-Lie algebra]].

If $\mathfrak{g}$ is concentrated in $\mathbb{Z}$-degrees 0 to $n-1$ then it is called a \emph{[[super Lie n-algebra]]}.

In particular if $\mathfrak{g}$ is concentrated in degree 0, then it is equivalently a [[super Lie algebra]].

Combining this, if $\mathfrak{g}$ is concentrated in even $\mathbb{Z}/2$-degree and in $\mathbb{Z}$-degree 0 through $n-1$, then it is called a \emph{[[Lie n-algebra]]}.

In particular if $\mathfrak{g}$ is concentrated in $\mathbb{Z}$-degree 0 and in even $\mathbb{Z}/2$-degree, then it is equivalently a plain [[Lie algebra]].

\end{remark}
\hypertarget{history}{}\subsection*{{History}}\label{history}

\begin{remark}
\label{SuperLInfintiyAsFDA}\hypertarget{SuperLInfintiyAsFDA}{}
\textbf{(history of the concept of (super-)$L_\infty$ algebras)}

The identification of the concept of (super-)$L_\infty$-algebras has a non-linear history:

[[L-∞ algebras]] in the incarnation of higher brackets satisfying a higher Jacobi identity, def. \ref{sLInfinityDefinitionViaGeneralizedJacobiIdentity} and remark \ref{LInfinityTerminology}, were introduced in \href{https://ncatlab.org/nlab/show/L-infinity-algebra#LadaStasheff92}{Lada-Stasheff 92}, based on the example of such a structure on the [[BRST complex]] of the [[bosonic string]] that was found in the construction of [[closed string field theory]] in \href{string+field+theory#Zwiebach93}{Zwiebach 92}. Some of this history is recalled in \hyperlink{L-infinity-algebra#Stasheff16}{Stasheff 16}.

The observation that these systems of higher brackets are fully characterized by their Chevalley-Eilenberg dg-(co-)algebras (def. \ref{SuperLInfinityCEAlgebra}) is due to \href{https://ncatlab.org/nlab/show/L-infinity-algebra#LadaMarkl94}{Lada-Markl 94}. See \hyperlink{SatiSchreiberStasheff08}{Sati-Schreiber-Stasheff 08, around def. 13}.

But in this dual incarnation, [[L-∞ algebras]] and more generally [[super L-∞ algebras]] (of [[finite type]]) had secretly been introduced within the [[supergravity]] literature already in \hyperlink{DAuriaFreRegge80}{D'Auria-Fr\'e{}-Regge 80} and explicitly in \hyperlink{Nieuwenhuizen82}{van Nieuwenhuizen 82}. The concept was picked up in the [[D'Auria-Fré formulation of supergravity]] (\hyperlink{DAuriaFre82}{D'Auria-Fr\'e{} 82}) and eventually came to be referred to as ``FDA''s (short for ``free differential algebra'') in the [[supergravity]] literature (but beware that these dg-algebras are in general [[free construction|free]] only as graded-[[supercommutative superalgebras]], not as differential algebras) The relation between super $L_\infty$-algebras and the ``FDA''s of the [[supergravity]] literature is made explicit in (\hyperlink{FSS13}{FSS 13}).

\begin{tabular}{l|l}
[[nLab:higher Lie theory]]&[[nLab:supergravity]]\\
\hline 
$\,$ [[nLab:super Lie n-algebra]] $\mathfrak{g}$ $\,$&$\,$ ``FDA'' $CE(\mathfrak{g})$ $\,$\\
\end{tabular}

The construction in \hyperlink{Nieuwenhuizen82}{van Nieuwenhuizen 82} in turn was motivated by the [[Sullivan algebras]] in [[rational homotopy theory]] (\hyperlink{rational+homotopy+theory#Sullivan77}{Sullivan 77}). Indeed, their dual incarnations in rational homotopy theory are [[dg-Lie algebras]] (\hyperlink{rational+homotopy+theory#Quillen69}{Quillen 69}), hence a special case of $L_\infty$-algebras (remark \ref{LInfinityTerminology})

This close relation between [[rational homotopy theory]] and [[higher Lie theory]] might have remained less of a secret had it not been for the focus of [[Sullivan minimal models]] on the non-[[simply connected topological space|simply connected]] case, which excludes the ordinary [[Lie algebras]] from the picture. But the Quillen model of [[rational homotopy theory]] effectively says that for $X$ a [[rational topological space]] then its [[loop space]] [[∞-group]] $\Omega X$ is reflected, infinitesimally, by an [[L-∞ algebra]]. This perspective began to receive more attention when the [[Sullivan construction]] in [[rational homotopy theory]] was concretely identified as higher [[Lie integration]] in \hyperlink{Lie+integration#Henriques}{Henriques 08}. A modern review that makes this [[L-∞ algebra]]-theoretic nature of [[rational homotopy theory]] manifest is in \href{rational+homotopy+theory#BuijsFelixMurillo12}{Buijs-F\'e{}lix-Murillo 12, section 2}.

\end{remark}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\begin{itemize}%
\item Every super $L_\infty$-Lie algebra has a [[Lie integration]] to a [[super ∞-groupoid]] and a [[smooth super ∞-groupoid]]. See at \emph{[[Lie integration]]} for more on this.

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

In the context of [[supergravity]]/[[string theory]] the

\begin{itemize}%
\item [[super Poincaré Lie algebra]]

\end{itemize}
and its super-$L_\infty$-extensions to the

\begin{itemize}%
\item [[supergravity Lie 3-algebra]] ([[m2brane]])


\item [[supergravity Lie 6-algebra]]


\item [[type II supergravity Lie 2-algebra]]



\end{itemize}
play a central role. Their exceptional [[infinity-Lie algebra cohomology]] governs the consistent [[Green-Schwarz action functionals]] for super-$p$-[[brane|branes]]. (See the discusson of the \emph{\href{Green-Schwarz+action+functional#BraneScan}{brane scan}}) there.

See at \emph{[[geometry of physics -- fundamental super p-branes]]} for more on this.

$\,$

The [[BRST complex]] of the [[superstring]] might form a super $L_\infty$-algebra whose brackets give the [[n-point function]] of the string, in analogy to what happens for the bosonic string in Zwiebach's [[string field theory]]. (\ldots{})

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

\begin{itemize}%
\item [[model structure on chain complexes of super vector spaces]]


\item [[model structure on differential graded-commutative superalgebras]]



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

In their [[formal duality|formal dual]] incarnations as super-graded commutative [[dg-algebras]] (super [[Chevalley-Eilenberg algebras|Chevalley-Eilenberg algebras]]), super $L_\infty$-algebras of [[finite type]] had secretly been introduced in

\begin{itemize}%
\item [[Riccardo D'Auria]], [[Pietro Fré]] [[Tullio Regge]], \emph{Graded Lie algebra, cohomology and supergravity}, Riv. Nuov. Cim. 3, fasc. 12 (1980) (\href{http://inspirehep.net/record/156191}{spire})


\item [[Peter van Nieuwenhuizen]], \emph{Free Graded Differential Superalgebras}, in \emph{Istanbul 1982, Proceedings, Group Theoretical Methods In Physics}, 228-247 and CERN Geneva - TH. 3499 (\href{http://inspirehep.net/record/182644/}{spire})



\end{itemize}
and hence a whole decade before the explicit appearance of plain (non-super) [[L-∞ algebras]] in \href{L-infinity-algebra#LadaStasheff92}{Lada-Stasheff 92}.

The concept was picked up in the [[D'Auria-Fré formulation of supergravity]]

\begin{itemize}%
\item [[Riccardo D'Auria]], [[Pietro Fré]], \emph{[[GeometricSupergravity.pdf:file]]}, Nuclear Physics B201 (1982) 101-140 ([[GeometricSupergravityErrata.pdf:file]])

\end{itemize}
and eventually came to be referred to as ``FDA''s (short for ``free differential algebra'') in the [[supergravity]] literature (where in [[rational homotopy theory]] one says ``[[semifree dga]]'', since these dg-algebras are crucially not required to be free as \emph{differential} algebras).

The relation between super $L_\infty$-algebras and the ``FDA''s of the [[supergravity]] literature is made explicit in

\begin{itemize}%
\item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], \emph{[[schreiber:The brane bouquet|Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields]]} (\href{https://arxiv.org/abs/1308.5264}{arXiv:1308.5264})

\end{itemize}
[[!redirects super L-∞ algebra]] [[!redirects super L-∞ algebras]]

[[!redirects super L-infinity algebras]]

[[!redirects super Lie n-algebra]] [[!redirects super Lie n-algebras]]

[[!redirects super Lie-infinity algebra]]

[[!redirects super Lie-∞ algebra]] [[!redirects super Lie-∞ algebras]]

[[!redirects FDA]] [[!redirects FDAs]]



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