81P05-LieAlgebroids.tex

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\begin{document}
\subsection{Topic on Lie algebroids}
 This is a topic entry on Lie algebroids that focuses on their quantum applications and extensions of current algebraic theories. 

\emph{Lie algebroids} generalize \emph{Lie algebras}, and in certain quantum systems they represent \emph{extended quantum (algebroid) symmetries}. One can think of a \emph{Lie algebroid} as generalizing the idea of a tangent bundle where the tangent space at a point is effectively the equivalence class of curves meeting at that point (thus
suggesting a groupoid approach), as well as serving as a site on which to study infinitesimal geometry (see, for example, ref. \cite{Mackenzie2005}). The formal definition of a \emph{Lie algebroid} is presented next. \\

\textbf{Definition 0.1}
Let $M$ be a manifold and let $\mathfrak X(M)$ denote the set of vector fields on $M$. Then, a 
\emph{Lie algebroid} over $M$ consists of a \emph{vector bundle $E \lra M$,
equipped with a Lie bracket $[~,~]$ on the space of sections $\gamma(E)$, 
and a bundle map $\Upsilon : E \lra TM$}, usually called the \emph{anchor}. 
Furthermore, there is an induced map $\Upsilon : \gamma (E) \lra \mathfrak X(M)$, 
which is required to be a map of Lie algebras, such that given sections $\a, \beta \in
\gamma(E)$ and a differentiable function $f$, the following
Leibniz rule is satisfied~:

\begin{equation}
[ \a, f \beta] = f [\a, \beta] + (\Upsilon (\a)) \beta~.
\end{equation}


\begin{example}  
A typical example of a Lie algebroid is obtained when $M$ is a Poisson
manifold and $E=T^*M$, that is $E$ is the cotangent bundle of $M$.
\end{example}

Now suppose we have a Lie groupoid $\mathsf{G}$:
\begin{equation}
r,s~:~ \xymatrix{ \mathsf{G} \ar@<1ex>[r]^r \ar[r]_s & \mathsf{G}^{(0)}}=M~.
\end{equation}
There is an associated Lie algebroid $\A = \A( \mathsf{G})$, which in the
guise of a vector bundle, it is the restriction to $M$ of the
bundle of tangent vectors along the fibers of $s$ (ie. the
$s$--vertical vector fields). Also, the space of sections $\gamma
(\A)$ can be identified with the space of $s$--vertical,
right--invariant vector fields $\mathfrak X^s_{inv} (\mathsf{G})$ which
can be seen to be closed under $[~,~]$, and the latter induces a
bracket operation on $\gamma (A)$ thus turning $\A$ into a Lie
algebroid. Subsequently, a Lie algebroid $\A$ is integrable if
there exists a Lie groupoid $\mathsf{G}$ inducing $\A$~.

\begin{remark}
Unlike Lie algebras that can be integrated to corresponding Lie groups, not all {\em Lie algebroids} are `smoothly integrable' to Lie groupoids; the subset of Lie groupoids that have corresponding Lie algebroids are sometimes called {\em `Weinstein groupoids'}.
\end{remark}

Note also the relation of the Lie algebroids to  Hamiltonian algebroids, also concerning recent developments in  (relativistic) quantum gravity theories.

\begin{thebibliography}{9}
\bibitem{Mackenzie2005}
K. C. H. Mackenzie: \emph{General Theory of Lie Groupoids and Lie
Algebroids}, London Math. Soc. Lecture Notes Series, \textbf{213},
Cambridge University Press: Cambridge,UK (2005).
\end{thebibliography}

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