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\section*{Lie algebroid}

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

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

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

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{in_terms_of_vector_bundles_with_anchor}{In terms of vector bundles with anchor}\dotfill \pageref*{in_terms_of_vector_bundles_with_anchor} \linebreak
\noindent\hyperlink{the_cealgebra_of_a_vector_bundle_with_anchor}{The CE-algebra of a vector bundle with anchor}\dotfill \pageref*{the_cealgebra_of_a_vector_bundle_with_anchor} \linebreak
\noindent\hyperlink{semifree_dgalgebras}{Semi-free dg-algebras}\dotfill \pageref*{semifree_dgalgebras} \linebreak
\noindent\hyperlink{lierinehart_algebras}{Lie-Rinehart algebras}\dotfill \pageref*{lierinehart_algebras} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{lie_theory_2}{Lie theory}\dotfill \pageref*{lie_theory_2} \linebreak
\noindent\hyperlink{poisson_geometry}{Poisson geometry}\dotfill \pageref*{poisson_geometry} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A \emph{Lie algebroid} is the [[horizontal categorification|many object version]] of a [[Lie algebra]]. It is the [[infinitesimal space|infinitesimal]] approximation to a [[Lie groupoid]].

There are various equivalent definitions:

\hypertarget{in_terms_of_vector_bundles_with_anchor}{}\subsubsection*{{In terms of vector bundles with anchor}}\label{in_terms_of_vector_bundles_with_anchor}

\begin{defn}
\label{}\hypertarget{}{}
A Lie algebroid over a [[manifold]] $X$ is

\begin{itemize}%
\item a [[vector bundle]] $E \to X$;


\item equipped with a Lie brackets $[\cdot,\cdot] : \Gamma(E)\otimes \Gamma(E) \to \Gamma(E)$ (over the ground field) on its space of sections;


\item a morphisms of vector bundles $\rho : E \to TX$, whose tangent map preserves the bracket: $(d\rho)([\xi,\zeta]_{\Gamma E}) = [d\rho(\xi),d\rho(\zeta)]_{\Gamma TX}$; (but this property of preserving brackets is implied by the next property, see Y. Kosmann-Schwarzbach and F. Magri. Poisson-Nijenhuis structures. Ann. Inst. H. Poinar\'e{} Phys. Th\'e{}or., 53(1):3581, 1990.)


\item such that the \emph{Leibniz rule} holds: for all $X, Y \in \Gamma(E)$ and all $f \in C^\infty(X)$ we have

\begin{displaymath}
[X, f \cdot Y] = f\cdot [X,Y] + \rho(X)(f) \cdot Y
\,.
\end{displaymath}


\end{itemize}
\end{defn}
\hypertarget{the_cealgebra_of_a_vector_bundle_with_anchor}{}\subsubsection*{{The CE-algebra of a vector bundle with anchor}}\label{the_cealgebra_of_a_vector_bundle_with_anchor}

Given this data of a vector bundle $E \to X$ with anchor map $\rho$ as above, one obtains the structure of a [[dg-algebra]] on the [[exterior algebra]] $\wedge^\bullet_{C^\infty(X)} \Gamma(E)^*$ of smooth sections of the dual bundle by the formula

\begin{displaymath}
(d\omega)(e_0, \cdots, e_n)
  = 
  \sum_{\sigma \in Shuff(1,n)} sgn(\sigma) \rho(e_{\sigma(0)})(\omega(e_{\sigma(1)}, \cdots, e_{\sigma(n)}))
  +
  \sum_{\sigma \in Shuff(2,n-1)} sign(\sigma) 
   \omega([e_{\sigma(0)},e_{\sigma(1)}],e_{\sigma(2)}, \cdots, e_{\sigma(n)} )
  \,,
\end{displaymath}
for all $\omega \in \wedge^n_{C^\infty(X)} \Gamma(E)^*$ and $(e_i \in \Gamma(E))$, where $Shuff(p,q)$ denotes the set of $(p,q)$-[[shuffle]]s $\sigma$ and $sgn(\sigma)$ the [[signature]] $\in \{\pm 1\}$ of the corresponding [[permutation]].

More details on this are at [[Chevalley-Eilenberg algebra]].

Conversely, one finds that every [[semi-free dga]] finitely generated in degree 1 over $C^\infty(X)$ arises this way, so that one may turn this around:

\hypertarget{semifree_dgalgebras}{}\subsubsection*{{Semi-free dg-algebras}}\label{semifree_dgalgebras}

\begin{defn}
\label{}\hypertarget{}{}
A Lie algebroid over a manifold $X$ is a vector bundle $E \to X$ equipped with a degree +1 derivation $d$ on the free (over $C^\infty(X)$) graded-commutative algebra $\wedge^\bullet_{C^\infty(X)} \Gamma(E)^*$ (where the dual is over $C^\infty$), such that $d^2 = 0$.

\end{defn}
This is for $\Gamma(E)$ satisfying suitable finiteness conditions. In general, as the masters well knew, the correct definition is the algebra of alternating multilinear functions from $\Gamma(E)$ to the ground field, assumed of characteristic 0. This can also be phrased in terms of linear maps from the corresponding coalgebra cogenerated by $\Gamma(E)$, but the masters did not have coalgebras in those days.

The differential graded-commutative algebra

\begin{displaymath}
CE(\mathfrak{g})
  :=
  (\wedge^\bullet_{C^\infty(X)} \Gamma(E)^*, d)
\end{displaymath}
is the [[Chevalley-Eilenberg algebra]] of the Lie algebroid (in that for $X = pt$ it reduces to the ordinary Chevally--Eilenberg algebra for Lie algebras).

In the existing literature this is often addressed just as ``the complex that computes Lie algebroid cohomology''.

It is helpful to compare this definition to the general definition of [[Lie infinity-algebroid|Lie ∞-algebroids]], the [[vertical categorification]] of Lie algebras and Lie algebroids.

\hypertarget{lierinehart_algebras}{}\subsubsection*{{Lie-Rinehart algebras}}\label{lierinehart_algebras}

\begin{defn}
\label{}\hypertarget{}{}
A Lie algebroid over the manifold $X$ is

\begin{itemize}%
\item a Lie algebra $\mathfrak{g}$;


\item the structure of a Lie module over $\mathfrak{g}$ on $C^\infty(X)$ (i.e. an action of $\mathfrak{g}$ on $X$);


\item the structure of a $C^\infty(X)$-module on $\mathfrak{g}$ (in fact: such that $\mathfrak{g}$ is a finitely generated projective module);


\item such that the two actions satisfy two compatibility conditions which are modeled on the standard relations obtained by setting $\mathfrak{g} = \Gamma(T X)$.



\end{itemize}
\end{defn}
This is the special case of a [[Lie-Rinehart pair]] $(A,\mathfrak{g})$ where the associative algebra $A$ is of the form $C^\infty(X)$.

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

\begin{itemize}%
\item A \textbf{[[Lie algebra]]} is a Lie algebroid over a point, $X = pt$.


\item The \textbf{[[tangent Lie algebroid]]} is

\begin{enumerate}%
\item in the vector bundle definition given by $E = T X$, $\rho = \mathrm{Id}$;


\item in the [[Chevalley-Eilenberg algebra]] definition: $\mathrm{CE}(T X) = (\Omega^\bullet(X), d_{deRham})$;



\end{enumerate}

\item An [[action Lie algebroid]] is the Lie version of an [[action groupoid]].


\item \textbf{Bundles of Lie algebras} $E \to X$ with fiber $\mathfrak{g}$ are Lie algebroids with $\rho = 0$ and fiberwise bracket. In particular, for $G$ a Lie group with Lie algebra $\mathfrak{g}$ and $P \to X$ a $G$-principal bundle, the \emph{adjoint bundle} $ad P := P \times_G \mathfrak{g}$ (where $\mathfrak{g}$ is associated using the [[adjoint representation]] of $G$ on its Lie algebra) is a bundle of Lie algebras.


\item Lie algebroids with injective anchor maps are equivalently [[integrable distributions]] in the [[tangent bundle]] of their base manifold and hence are equivalently [[foliations]] of their base manifold.


\item The \textbf{[[Atiyah Lie algebroid]]} is the Lie algebroid of the [[Atiyah Lie groupoid]] of a principal bundle: for $G$ a Lie group and $P \to X$ a $G$-principal bundle, the vector bundle $At(P):= T P/G$ naturally inherits the structure of a Lie algebroid. Moreover, it fits into a short exact sequence of Lie algebroids over $X$

\begin{displaymath}
0 \to ad P \to At(P) \to T X \to 0
\end{displaymath}
known as the \textbf{Atiyah sequence}.


\item The \textbf{vertical tangent Lie algebroid} $T_{vert}Y \hookrightarrow T Y$ of a smooth map $\pi : Y \to X$ of manifolds is the sub-Lie algebroid of the tangent Lie algebroid $T Y$ defined as follows:

\begin{enumerate}%
\item In the vector bundle perspective $E = ker(\pi_*)$ is the kernel bundle of the map $\pi_* : T Y \to T X$.


\item In the dual picture we have $CE(T_{vert}Y) = \Omega^\bullet_{vert}(Y)$, the [[Lie infinity-algebroid|qDGCA]] of \textbf{vertical differential forms}. This is the quotient of $\Omega^\bullet(Y)$ by the ideal of those forms which vanish when restricted in all arguments to $ker(\pi_*)$.



\end{enumerate}

\item Each [[Poisson manifold]] $(X,\pi)$ defines and is defined by a [[Poisson Lie algebroid]] $T^* X \stackrel{\pi}{\to} t X$. This is the degree-1 example of a more general structure described at [[n-symplectic manifold]].


\item If $E\to X$ is a Lie algebroid with bracket $[,]$ and anchor $\rho:E\to TX$ then it induces a Lie algebroid structure on the $k$-th [[jet bundle|jet]] bundle $j^k E\to X$, called the \textbf{jet Lie algebroid}. More precisely, if $s\in\Gamma_X E$ then call by $j^k s$ the induced section in $\Gamma_X j^k E$. Then there is a unique Lie algebroid structure on the bundle $j^k E\to X$ such that the following two properties hold: $[j^k s, j^k t] = j^k [s,t]$ and $\rho(j^k s) = \rho(s)$ for all $s,t\in\Gamma_X E$ (see \href{http://www.math.ist.utl.pt/~rfern/Meus-papers/HTML/jets.pdf}{pdf}).


\item A [[BRST complex]] is a [[Chevalley-Eilenberg algebra]] of a Lie algebroid which corresponds to the [[action groupoid]] of a Lie group acting on a space.



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

\hypertarget{lie_theory_2}{}\subsubsection*{{Lie theory}}\label{lie_theory_2}

The extent to which Lie algebroids are to [[Lie groupoid|Lie groupoids]] as Lie algebras are to Lie groups is the content of general [[Lie theory]], in which [[Lie's three theorems|Lie's theorems]] have been generalized to Lie algebroids.

\hypertarget{poisson_geometry}{}\subsubsection*{{Poisson geometry}}\label{poisson_geometry}

The fiberwiese linear dual of a Lie algebroid (regarded as a vector bundle) is naturally a [[Poisson manifold]]: the \emph{[[Lie-Poisson structure]]}.

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

\begin{itemize}%
\item [[Lie algebra]], \textbf{Lie algebroid}


\item [[double Lie algebroid]]


\item [[L-∞ algebra]], [[L-∞ algebroid]]


\item [[foliation of a Lie algebroid]]


\item [[Lie algebroid cohomology]]


\item [[super Lie algebroid]]



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

The concept of Lie algebroid was introduced in

\begin{itemize}%
\item [[Jean Pradines]], \emph{Th\'e{}orie de Lie pour les groupo\"i{}des diff\'e{}rentiables. Calcul diff\'e{}renetiel dans la cat\'e{}gorie des groupo\"i{}des infinit\'e{}simaux}, C. R. Acad. Sci. Paris S\'e{}r. A-B 264 1967 A245--A248, \href{http://www.ams.org/mathscinet-getitem?mr=0216409}{MR0216409}

\end{itemize}
In algebra a generalization of Lie algebroid, the Lie pseduoalgebra or Lie-Rinehart algebra/pair has been introduced more than a dozen of times under various names starting in early 1950-s. Atiyah's construction of Atiyah sequence is published in 1957 and Rinehart's paper in 1963.

Historically important is also the reference

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

\end{itemize}
on [[tangent Lie algebroid]]s.

Contemporary textbooks include:

\begin{itemize}%
\item K. C. H. Mackenzie, \emph{General theory of Lie groupoids and Lie algebroids,} Cambridge University Press, 2005, xxxviii + 501 pages (\href{http://kchmackenzie.staff.shef.ac.uk/gt.html}{website})


\item K. C. H. Mackenzie, \emph{Lie groupoids and Lie algebroids in differential geometry}, London Mathematical Society Lecture Note Series, 124. Cambridge University Press, Cambridge, 1987. xvi+327 pp (\href{http://www.ams.org/mathscinet-getitem?mr=896907}{MathSciNet})


\item [[Janez Mrčun]], [[Ieke Moerdijk]], \emph{Introduction to foliations and Lie groupoids}, Cambridge Studies in Advanced Mathematics \textbf{91}, Cambridge University Press 2003. x+173 pp. ISBN: 0-521-83197-0



\end{itemize}
For an infinite-dimensional version used in stochastic analysis see

\begin{itemize}%
\item R\'e{}mi L\'e{}andre, \emph{A Lie algebroid on the Wiener space}, Adv. Math. Phys. 2010, Art. ID 146719, 17 pp. \href{http://www.ams.org/mathscinet-getitem?mr=2594930}{MR2011j:58064}

\end{itemize}
There is also a recent ``hom-version''

\begin{itemize}%
\item Camille Laurent-Gengoux, Joana Teles, \emph{Hom-Lie algebroids}, \href{http://arxiv.org/abs/1211.2263}{arxiv/1211.2263}

\end{itemize}
[[!redirects Lie algebroids]]

[[!redirects anchor map]] [[!redirects anchor maps]]



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