\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*{Poisson Lie 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{as_vectorbundle_with_anchor}{As vector-bundle with anchor}\dotfill \pageref*{as_vectorbundle_with_anchor} \linebreak
\noindent\hyperlink{chevalleyeilenberg_algebra}{Chevalley-Eilenberg algebra}\dotfill \pageref*{chevalleyeilenberg_algebra} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{Cohomology}{Cohomology and Chern-Simons elements}\dotfill \pageref*{Cohomology} \linebreak
\noindent\hyperlink{lagrangian_submanifolds_and_coisotropic_submanifolds}{Lagrangian submanifolds and coisotropic submanifolds}\dotfill \pageref*{lagrangian_submanifolds_and_coisotropic_submanifolds} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A Poisson Lie algebroid on a [[manifold]] $X$ is a [[Lie algebroid]] on $X$ naturally defined from and defining the structure of a [[Poisson manifold]] on $X$.

This is the degree-1 example of a tower of related concepts, described at [[n-symplectic manifold]].

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

Let $\pi \in \Gamma(\Wedge^2 TX)$ be a [[Poisson manifold]] structure, incarnated as a [[Poisson tensor]].

\hypertarget{as_vectorbundle_with_anchor}{}\subsubsection*{{As vector-bundle with anchor}}\label{as_vectorbundle_with_anchor}

In terms of the vector-bundle-with anchor definition of [[Lie algebroid]] the \textbf{Poisson Lie algebroid} $\mathfrak{P}(X,\pi)$ corresponding to $\pi$ is the [[cotangent bundle]]

\begin{displaymath}
\itexarray{
      T^* X &&\stackrel{\pi(-)}{\to}&& T X
      \\
      & \searrow && \swarrow
      \\
      && X
  }
\end{displaymath}
equipped with the anchor map that sends a [[differential 1-form]] $\alpha$ to the [[vector]] obtained by contraction with the [[Poisson bivector]] $\pi \colon \alpha \mapsto \pi(\alpha,-)$.

The [[Lie bracket]] $[-,-] : \Gamma(T^* X) \wedge \Gamma(T^* X) \to \Gamma(T^* X)$ is given by

\begin{displaymath}
[\alpha,\beta] 
   \coloneqq 
  \mathcal{L}_{\pi(\alpha)} \beta 
  - \mathcal{L}_{\pi(\beta)} \alpha 
  - d_{dR}(\pi(\alpha,\beta))\,,
\end{displaymath}
where $\mathcal{L}$ denotes the [[Lie derivative]] and $d_{dR}$ the [[de Rham differential]]. This is the unique Lie algebroid bracket on $T^* X \stackrel{\pi}{\to} T X$ which is given on exact differential 1-forms by

\begin{displaymath}
[d_{dR} f, d_{dR} g] = d_{dR} \{f,g\}
\end{displaymath}
for all $f,g \in C^\infty(X)$. On a [[coordinate patch]] this reduces to

\begin{displaymath}
[d x^i , d x^j] = d_{dR} \pi^{i j}
\end{displaymath}
for $\{x^i\}$ the coordinate functios and $\{\pi^{i j}\}$ the components of the [[Poisson tensor]] in these coordinates.

\hypertarget{chevalleyeilenberg_algebra}{}\subsubsection*{{Chevalley-Eilenberg algebra}}\label{chevalleyeilenberg_algebra}

We describe the [[Chevalley-Eilenberg algebra]] of the Poisson Lie algebra given by $\pi$, which defines it dually.

Notice that $\pi$ is an element of degree 2 in the [[exterior algebra]] $\wedge^\bullet \Gamma(T X)$ of [[multivector field]]s on $X$. The Lie bracket on [[tangent bundle|tangent vector]]s in $\Gamma(T X)$ extends to a bracket $[-,-]_{Sch}$ on multivector field, the \textbf{[[Schouten bracket]]}. The defining property of the Poisson structure $\pi$ is that

\begin{displaymath}
[\pi,\pi]_{Sch} = 0
  \,.
\end{displaymath}
This makes

\begin{displaymath}
d_{CE(\mathfrak{P}(X,\pi))} := [\pi, -] : CE(\mathfrak{P}(X,\pi)) \to CE(\mathfrak{P}(X,\pi)))
\end{displaymath}
into a differential of degree +1 on multivector fields, that squares to 0. We write $CE(\mathfrak{P}(X,\pi))$ for the exterior algebra equipped with this differential.

More explicitly, let $\{x^i\} : X \to \mathbb{R}^{dim X}$ be a coordinate patch. Then the differential of $CE(\mathfrak{P}(X,\pi))$ is given by

\begin{displaymath}
d_{\mathfrak{P}(X,\pi)} : x^i \mapsto  2 \pi^{i j} \partial_j
\end{displaymath}
\begin{displaymath}
d_{\mathfrak{P}(X,\pi)} : \partial_i \mapsto ...
  \,.
\end{displaymath}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{Cohomology}{}\subsubsection*{{Cohomology and Chern-Simons elements}}\label{Cohomology}

We discuss aspects of the [[∞-Lie algebra cohomology|∞-Lie algebroid cohomology]] of Poisson Lie algebroids $\mathfrak{P}(X,\pi)$. This is equivalently called \emph{[[Poisson cohomology]]} (see there for details).

We shall be lazy (and follow tradition) and write the following formulas in a local coordinate patch $\{x^i\}$ for $X$.

Then the [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{P}(X,\pi))$ is generated from the $x^i$ and the $\partial_i$, and the [[Weil algebra]] $W(\mathfrak{P}(X,\pi))$ is generated from $x^i$, $\partial_i$ and their shifted partners, which we shall write $\mathbf{d} x^i$ and $\mathbf{d}\partial_i$. The differential on the Weil algebra we may then write

\begin{displaymath}
d_{W(\mathfrak{P}(X,\pi))} = [\pi,-]_{Sch} + \mathbf{d}
  \,.
\end{displaymath}
Notice that $\pi \in CE(\mathfrak{P}(X,\pi))$ is a [[∞-Lie algebra cohomology|Lie algebroid cocycle]], since

\begin{displaymath}
d_{CE(\mathfrak{P}(X,\pi))} \pi = [\pi,\pi]_{Sch} = 0
  \,.
\end{displaymath}
\begin{prop}
\label{}\hypertarget{}{}
The [[invariant polynomial]] in transgression with $\pi$ is

\begin{displaymath}
\omega =  (\mathbf{d}\partial_i) \wedge (\mathbf{d}x^i)
   \in
  W(\mathfrak{P}(X,\pi))
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
One checks that the following is a \textbf{[[Chern-Simons element]]} (see there for more) exhibiting the transgression

\begin{displaymath}
cs_\pi = \pi^{i j} \partial_i  \wedge \partial_j 
  + \partial_i \wedge \mathbf{d}x^i
  \;\;\;
  \in W(\mathfrak{P}(X,\pi))
\end{displaymath}
in that $d_{W(\mathfrak{P}(X,\pi))} cs_\pi = \omega$, and the restriction of $cs_\pi$ to $CE(\mathfrak{P}(X,\pi))$ is evidently the Poisson tensor $\pi$.

For the record (and for the signs) here is the explicit computation

\begin{displaymath}
\begin{aligned}
    d_{W(\mathfrak{P}(X,\pi))} (\pi^{i j} \partial_i \wedge \partial_j
   + \partial_i \wedge \mathbf{d} x^i)
    = &
   \mathbf{d}x^k (\partial_k \pi^{i j}) \partial_i \wedge \partial_j
   \\
   & + 2 \pi^{i j} (\mathbf{d}\partial_i) \wedge \partial_j
   \\
   & - (\partial_i \pi^{j k}) \partial_j \wedge \partial_k \wedge 
       \mathbf{d}x^i
   \\
   & +
   (\mathbf{d}\partial_i)\wedge (\mathbf{d} x^i)
   \\
   & + (-)(-) 2\pi^{i j} \partial_i \wedge \mathbf{d}\partial_j
   \\
   = & 
   (\mathbf{d}\partial_i)\wedge (\mathbf{d} x^i)
  \end{aligned}
  \,.
\end{displaymath}
\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
The invariant polynomial $\omega$ makes $\mathfrak{P}(X,\pi)$ a [[schreiber:symplectic ∞-Lie algebroid]].

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
The [[schreiber:infinity-Chern-Simons theory]] [[action functional]] induced from the above Chern-Simons element is that of the [[Poisson sigma-model]]:

it sends [[∞-Lie algebroid valued forms]]

\begin{displaymath}
\Omega^\bullet(\Sigma) \leftarrow W(\mathfrak{P}(X,\pi))
  (X,\eta)
\end{displaymath}
on a 2-dimensional manifold $\Sigma$ with values in a Poisson Lie algebroid on $X$ to the integral of the [[Chern-Simons form|Chern-Simons 2-form]]

\begin{displaymath}
\Omega^\bullet(\Sigma) \leftarrow 
  W(\mathfrak{P}(X,\pi))
  \stackrel{(\omega, cs_\omega)}{\leftarrow}
  W(b^2 \mathbb{R})
  : 
  CS_\omega(X,\eta)
\end{displaymath}
which, by the above, is in components

\begin{displaymath}
CS_\omega(X,\eta) = \eta_i \wedge d_{dR} X^i + \pi^{i j} \eta_i \wedge \eta_j
  \,.
\end{displaymath}
\end{remark}
\hypertarget{lagrangian_submanifolds_and_coisotropic_submanifolds}{}\subsubsection*{{Lagrangian submanifolds and coisotropic submanifolds}}\label{lagrangian_submanifolds_and_coisotropic_submanifolds}

The [[Lagrangian dg-submanifolds]] (see there for more) of a Poisson Lie algebroid correspond to the [[coisotropic submanifolds]] of the corresponding [[Poisson manifold]].

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

\begin{itemize}%
\item Under [[Lie integration]] a Poisson Lie algebroid is supposed to yield a [[symplectic groupoid]].


\item There is a formulation of [[Legendre transformation]] in terms of Lie algebroid.


\item [[symplectic Lie n-algebroid]]

\begin{itemize}%
\item [[symplectic manifold]]


\item \textbf{Poisson Lie algebroid}


\item [[Courant algebroid]]



\end{itemize}

\item [[Hopf algebroid]] (appears as a deformation quantization of a Poisson-Lie algebroid)



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

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

\begin{itemize}%
\item For the simple but important special case of [[Lie-Poisson structure]] see there at \emph{\href{Lie-Poisson+structure#PoissonLieAlgebroidCohomology}{Poisson-Lie algebroid cohomology}}.

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

One of the earliest reference seems to be

\begin{itemize}%
\item [[Ted Courant]], \emph{Tangent Lie algebroid} (\href{http://www.iop.org/EJ/article/0305-4470/27/13/026/ja941326.pdf}{pdf})

\end{itemize}
A review is for instance in

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

\end{itemize}
The [[H-cohomology]] of the graded symplectic form of a Poisson Lie algebroid, regarded a a [[symplectic Lie n-algebroid]], 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 Poisson-Lie algebroid]]

[[!redirects Poisson Lie algebroids]]



\end{document}