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\begin{document}

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\section*{symplectic Lie n-algebroid}

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

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

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

\hypertarget{symplectic_geometry}{}\paragraph*{{Symplectic geometry}}\label{symplectic_geometry}

[[!include symplectic geometry - 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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{_symplectic_manifold}{$n=0$: symplectic manifold}\dotfill \pageref*{_symplectic_manifold} \linebreak
\noindent\hyperlink{_poisson_manifold}{$n=1$: Poisson manifold}\dotfill \pageref*{_poisson_manifold} \linebreak
\noindent\hyperlink{_courant_algebroid}{$n=2$: Courant algebroid}\dotfill \pageref*{_courant_algebroid} \linebreak
\noindent\hyperlink{relation_to_other_concepts}{Relation to other concepts}\dotfill \pageref*{relation_to_other_concepts} \linebreak
\noindent\hyperlink{to_chernsimons_theory}{To $\infty$-Chern-Simons theory}\dotfill \pageref*{to_chernsimons_theory} \linebreak
\noindent\hyperlink{to_multisymplectic_geometry}{To multisymplectic geometry}\dotfill \pageref*{to_multisymplectic_geometry} \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 [[Lie n-algebroid]] is \emph{symplectic} if it is equipped with a non-degenerate binary [[invariant polynomial]]. This generalizes the notion of a [[symplectic form]] on a [[symplectic manifold]], to which it reduces for $n = 0$.

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

\begin{defn}
\label{}\hypertarget{}{}
A \textbf{symplectic Lie $n$-algebroid} is a pair

\begin{displaymath}
(\mathfrak{a}, \langle -,- \rangle)
\end{displaymath}
consisting of

\begin{itemize}%
\item a [[Lie n-algebroid]] $\mathfrak{a}$;


\item a binary [[invariant polynomial]] $\langle- , - \rangle$ of degree $(n+2)$

(a closed element in the shifted elements of the [[Weil algebra]] $W(\mathfrak{a})$)

which is non-degenerate.



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

The [[Chern-Simons element]] that witnesses this transgression is the Lagrangian of the corresponding [[AKSZ theory]] [[sigma-model]] with $\mathfrak{a}$ as its target space and the invariant polynomial $\langle -,- \rangle$ as the ([[curvature]] of) its background [[gauge field]].

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

\hypertarget{_symplectic_manifold}{}\subsubsection*{{$n=0$: symplectic manifold}}\label{_symplectic_manifold}

A 0-Lie algebroid is just a [[smooth manifold]] $X$.

Its [[Chevalley-Eilenberg algebra]] is the algebra of smooth functions on $X$

\begin{displaymath}
CE(X) = C^\infty(X)
  \,.
\end{displaymath}
The [[Weil algebra]] of $X$ is

\begin{displaymath}
W(X) = \Omega^\bullet(X)
\end{displaymath}
the [[de Rham complex|de Rham algebra]] of $X$. A degree 2-[[invariant polynomial]] on $X$ is therefore a non-degenerate closed [[differential form|2-form]] $\omega \in \Omega^2(X)$, a [[symplectic manifold|symplectic 2-form]].

A [[symplectic manifold]], being a pair

\begin{displaymath}
(X,\;\; \omega)
\end{displaymath}
consisting of a [[smooth manifold]] $X$ and a symplectic 2-form $\omega$, is a symplectic Lie 0-algebroid.

\hypertarget{_poisson_manifold}{}\subsubsection*{{$n=1$: Poisson manifold}}\label{_poisson_manifold}

For a [[Poisson manifold]] $X$ with Poisson bivector $\pi \in \Gamma(T X) \wedge \Gamma(T X)$ the [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{a})$ of the corresponding [[Poisson Lie algebroid]]

\begin{displaymath}
\mathfrak{a} := \mathfrak{P}(X,\pi)
\end{displaymath}
is that of multi-vector fields on $X$, equipped with the differential $d_{CE(\mathfrak{a})} = [\pi, -]_{Sch}$ given by the [[Schouten bracket]].

If we work locally in coordinates then $CE(\mathfrak{a})$ is generated from degree 0 elements $x^i$ and degree 1 elements $\partial_i$. The differential is

\begin{displaymath}
d_{CE(\mathfrak{a})} = [\pi, -]_{Sch}
  \,.
\end{displaymath}
The Poisson tensor is $\nu := \pi = \pi^{i j} \partial_i \wedge \partial_j$ and that this is a [[∞-Lie algebra cohomology|Lie algebroid cocycle]] is the fact that

\begin{displaymath}
d_{CE(\mathfrak{a})} \pi = [\pi,\pi]_{Sch} = 0
  \,.
\end{displaymath}
By definition the [[Weil algebra]] $W(\mathfrak{a})$ is generated from the $x^i$, the $\partial_i$ and their shifted partners $\mathbf{d}x^i$ and $\mathbf{d}\partial_i$. The differential here is

\begin{displaymath}
d_{W(\mathfrak{a})} = [\pi , - ] + \mathbf{d}
  \,.
\end{displaymath}
\begin{uprop}
The [[invariant polynomial]] $\omega$ that is in transgression with the cocycle $\nu = \pi$ is

\begin{displaymath}
\omega = (\mathbf{d} x^i) \wedge (\mathbf{d} \partial_i)
  \;\;\;
  \in inv(\mathfrak{a})
  \,.
\end{displaymath}
\end{uprop}
\begin{proof}
One checks directly that the element

\begin{displaymath}
cs_\omega = 
  \pi^{i j} \partial_i \wedge \partial_j
  +
  x^i \wedge \mathbf{d} \partial_i
\end{displaymath}
is a Chern-Simons transgression element for $\nu$ and $\omega$,

i.e. $d_{W(\mathfrak{a})} cs(\omega) = \omega$. The restriction of $cs_\omega$ to $CE(\mathfrak{a})$ is evidently the Poisson tensor $\pi$.

\end{proof}
More details on this at [[Chern-Simons element]].

For a [[Poisson manifold]] $X$ with Poisson tensor $\pi = \pi^{i j} \partial_i \wedge \partial_j$, the pair

\begin{displaymath}
(\mathfrak{P}(X,\pi), \;\;\;
\omega = (\mathbf{d} x^i) \wedge (\mathbf{d} \partial_i))
\end{displaymath}
consisting of the [[Poisson Lie algebroid]] $\mathfrak{P}(X,\pi)$ and of the [[invariant polynomial]] $\omega$ that is in transgression with its canonical 2-cocycle $\nu = \pi$ (the Poisson tensor) is a symplectic [[Lie algebroid]].

\hypertarget{_courant_algebroid}{}\subsubsection*{{$n=2$: Courant algebroid}}\label{_courant_algebroid}

A $2$-symplectic manifold encodes and is encoded by the structure of a [[Courant algebroid]].

A Courant 2algebroid over the point if given by a [[semisimple Lie algebra]] with the symplectic form being the [[Killing form]]. The coresponding Poisson tensor is the canonical 3-cocycle $\langle -, [-,-] \rangle$ on a semisimple Lie algebra. The extension classified by this is the [[string Lie 2-algebra]].

\hypertarget{relation_to_other_concepts}{}\subsection*{{Relation to other concepts}}\label{relation_to_other_concepts}

\hypertarget{to_chernsimons_theory}{}\subsubsection*{{To $\infty$-Chern-Simons theory}}\label{to_chernsimons_theory}

Since the symplectic form on a symplectic Lie $n$-Algebroid may be understood [[Lie theory|Lie theoretically]] as an [[invariant polynomial]] on an [[L-∞ algebroid]], every symplectic Lie $n$-algebroid serves as a [[target space]] for an [[schreiber:∞-Chern-Simons theory]]: this is [[AKSZ theory]].

We have

\begin{itemize}%
\item for $n = 1$ the [[Poisson sigma-model]];
\item for $n = 2$ the [[Courant sigma model]]
\item and so on.

\end{itemize}
\hypertarget{to_multisymplectic_geometry}{}\subsubsection*{{To multisymplectic geometry}}\label{to_multisymplectic_geometry}

There is also the closely related notion of [[multisymplectic geometry]]. See

\begin{itemize}%
\item [[John Baez]], [[Chris Rogers]], \emph{Categorified Symplectic Geometry and the String Lie 2-Algebra}, (\href{http://arxiv.org/abs/0901.4721}{arXiv})

\end{itemize}
for some relations of this to the above situation for $n = 2$. Essentially multisymplectic geometry studies the higher $n$-ary brackets induced from the binary graded symplectic form discussed here. The relation between these two pictures is the same as that between as studied in the context of [[hemistrict Lie 2-algebra]]s.

An article with more details on this:

\begin{itemize}%
\item [[Chris Rogers]], \emph{Courant algebroids from categorified symplectic geometry} (\href{http://math.ucr.edu/~chris/2plectic-algebroid_DRAFT.pdf}{pdf}).

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

\begin{itemize}%
\item [[L-∞ algebroid]]


\item [[symplectic groupoid]]



\end{itemize}
[[!include infinity-CS theory for binary non-degenerate invariant polynomial - table]]

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

The notion originates somewhere in the school of [[Alan Weinstein]]`s school of [[higher category theory|higher categorial]] [[symplectic geometry]]. The first published appearance of the notion at least for $0 \leq n \leq 3$ is

\begin{itemize}%
\item [[Dmitry Roytenberg]], \emph{Courant algebroids, derived brackets and even symplectic supermanifolds} PhD thesis (\href{http://arxiv.org/abs/math/9910078}{arXiv:9910078})

\end{itemize}
A good writeup of this material is in

\begin{itemize}%
\item [[Dmitry Roytenberg]], \emph{On the structure of graded symplectic supermanifolds and Courant algebroids} in \emph{Quantization, Poisson Brackets and Beyond} , [[Theodore Voronov]] (ed.), Contemp. Math., Vol. 315, Amer. Math. Soc., Providence, RI, 2002 (\href{http://arxiv.org/abs/math/0203110}{arXiv})

\end{itemize}
The idea for all $n$ was then sketched, together with many other ideas about [[L-infinity algebroid]]s in the article with the nice title

\begin{itemize}%
\item [[Pavol ?evera]], \emph{Some title containing the words ``homotopy'' and ``symplectic'', e.g. this one} (\href{http://arxiv.org/abs/math/0105080}{arXiv})

\end{itemize}
What we call $n$-symplectic manifold here is called $\Sigma_n$-manifold there.

\textbf{Warning} This article here uses the term ``$n$-symplectic'' in a related but not identical sense to the one used here:

\begin{itemize}%
\item M. de Leon, D. Martin de Diego, M. Salgado, S. Vilari\~n{}o, \emph{K-symplectic formalism on Lie algebroids} (\href{http://lanl.arxiv.org/abs/0905.4585}{arXiv})

\end{itemize}
A discussion of aspects of how [[multisymplectic geometry]] related to $n$-symplectic manifolds is in

\begin{itemize}%
\item [[Chris Rogers]], \emph{Courant algebroids from categorified symplectic geometry} (\href{http://math.ucr.edu/~chris/2plectic-algebroid_DRAFT.pdf}{pdf})

\end{itemize}


\begin{itemize}%
\item [[Chris Rogers]], \emph{Higher symplectic geometry} PhD thesis (2011) (\href{http://arxiv.org/abs/1106.4068}{arXiv:1106.4068})

\end{itemize}
A discussion of symplectic Lie n-algebroids from an [[infinity-Lie theory]] perspective as discussed here is in

\begin{itemize}%
\item [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], \emph{[[schreiber:AKSZ Sigma-Models in Higher Chern-Weil Theory]]}, Int. J. Geom. Methods Mod. Phys. 10 (2013) 1250078 (\href{http://arxiv.org/abs/1108.4378}{arXiv:1108.4378})

\end{itemize}
The [[H-cohomology]] of graded symplectic forms is considered in

\begin{itemize}%
\item [[Pavol ?evera]], p. 1 of \emph{On the origin of the BV operator on odd symplectic supermanifolds}, Lett Math Phys (2006) 78: 55. (\href{https://arxiv.org/abs/math/0506331}{arXiv:0506331})

\end{itemize}
[[!redirects symplectic Lie n-algebroids]]

[[!redirects n-symplectic manifold]]



\end{document}