Editing Lie algebroid (section)

==Definition and basic concepts==
A '''Lie algebroid''' is a triple <math>(A, [\cdot,\cdot], \rho)</math> consisting of

* a [[vector bundle]] <math>A</math> over a [[manifold]] <math>M</math>
* a [[Lie algebra#Definitions|Lie bracket]] <math>[\cdot,\cdot]</math> on its space of sections <math>\Gamma (A)</math>
* a morphism of vector bundles <math>\rho: A\rightarrow TM</math>, called the '''anchor''', where <math>TM</math> is the [[tangent bundle]] of <math>M</math>

such that the anchor and the bracket satisfy the following Leibniz rule:

:<math>[X,fY]=\rho(X)f\cdot Y + f[X,Y]</math>

where <math>X,Y \in \Gamma(A), f\in C^\infty(M)</math>. Here <math>\rho(X)f</math> is the image of <math>f</math> via the [[Derivation (differential algebra)|derivation]] <math>\rho(X)</math>, i.e. the [[Lie derivative#The (Lie) derivative of a function|Lie derivative]] of <math>f</math> along the vector field <math>\rho(X)</math>. The notation <math>\rho(X)f \cdot Y</math> denotes the (point-wise) product between the function <math>\rho(X)f</math> and the vector field <math>Y</math>.

One often writes ''<math>A \to M</math>'' when the bracket and the anchor are clear from the context; some authors denote Lie algebroids by ''<math>A \Rightarrow M</math>'', suggesting a "limit" of a Lie groupoids when the arrows denoting source and target become "infinitesimally close".<ref>{{Cite arXiv|last=Meinrenken|first=Eckhard|date=2021-05-08|title=On the integration of transitive Lie algebroids|class=math.DG|eprint=2007.07120}}</ref>

=== First properties ===
It follows from the definition that

* for every <math>x \in M</math>, the kernel <math>\mathfrak{g}_x(A)=\ker(\rho_x)</math> is a Lie algebra, called the '''isotropy Lie algebra''' at <math>x</math>
*the kernel <math>\mathfrak{g}(A)=\ker(\rho)</math> is a (not necessarily locally trivial) bundle of Lie algebras, called the '''isotropy Lie algebra bundle'''
* the image <math>\mathrm{Im}(\rho) \subseteq TM</math> is a [[Singular distribution (differential geometry)|singular distribution]] which is integrable, i.e. its admits maximal immersed submanifolds <math>\mathcal O \subseteq M</math>, called the '''orbits''', satisfying <math>\mathrm{Im}(\rho_x) = T_x \mathcal{O}</math> for every <math>x \in \mathcal O</math>. Equivalently, orbits can be explicitly described as the sets of points which are joined by '''A-paths''', i.e. pairs <math>(a: I \to A, \gamma: I \to M)</math> of paths in <math>A</math> and in <math>M</math> such that <math>a(t) \in A_{\gamma(t)}</math> and <math>\rho (a(t)) = \gamma'(t)</math>
* the anchor map <math>\rho</math> descends to a map between sections <math>\rho: \Gamma(A) \rightarrow \mathfrak{X}(M)</math> which is a Lie algebra morphism, i.e.
:<math>\rho([X,Y])=[\rho(X),\rho(Y)] </math>

for all <math>X,Y \in \Gamma(A)</math>.

The property that <math>\rho</math> induces a Lie algebra morphism was taken as an axiom in the original definition of Lie algebroid.<ref name=":0" /> Such redundancy, despite being known from an algebraic point of view already before Pradine's definition,<ref>{{Cite journal|last=J. C.|first=Herz|date=1953|title=Pseudo-algèbres de Lie|journal=C. R. Acad. Sci. Paris|language=fr|volume=236|pages=1935–1937}}</ref> was noticed only much later.<ref>{{Cite journal|last1=Kosmann-Schwarzbach|first1=Yvette|last2=Magri|first2=Franco|date=1990|title=Poisson-Nijenhuis structures|url=http://www.numdam.org/item/AIHPA_1990__53_1_35_0/|journal=Annales de l'Institut Henri Poincaré A|volume=53|issue=1|pages=35–81}}</ref><ref>{{Cite journal|last=Grabowski|first=Janusz|date=2003-12-01|title=Quasi-derivations and QD-algebroids|url=https://www.sciencedirect.com/science/article/pii/S0034487703800411|journal=Reports on Mathematical Physics|language=en|volume=52|issue=3|pages=445–451|doi=10.1016/S0034-4877(03)80041-1|issn=0034-4877|arxiv=math/0301234|bibcode=2003RpMP...52..445G|s2cid=119580956}}</ref>

=== Subalgebroids and ideals ===
A '''Lie subalgebroid''' of a Lie algebroid <math>(A, [\cdot,\cdot], \rho)</math> is a vector subbundle <math>A'\to M'</math> of the restriction <math>A_{\mid M'} \to M'</math> such that <math>\rho_{\mid A'}</math> takes values in <math>TM'</math> and <math>\Gamma(A,A'):= \{ \alpha \in \Gamma(A) \mid \alpha_{\mid M'} \in \Gamma(A') \}</math> is a Lie subalgebra of <math>\Gamma(A)</math>. Clearly, <math>A'\to M'</math> admits a unique Lie algebroid structure such that <math>\Gamma(A,A') \to \Gamma(A')</math> is a Lie algebra morphism. With the language introduced below, the inclusion <math>A' \hookrightarrow A</math> is a Lie algebroid morphism.

A Lie subalgebroid is called '''wide''' if <math>M' = M</math>. In analogy to the standard definition for Lie algebra, an '''ideal''' of a Lie algebroid is wide Lie subalgebroid <math>I \subseteq A</math> such that <math>\Gamma(I) \subseteq \Gamma(A)</math> is a Lie ideal. Such notion proved to be very restrictive, since <math>I</math> is forced to be inside the isotropy bundle <math>\ker(\rho)</math>. For this reason, the more flexible notion of '''infinitesimal ideal system''' has been introduced.<ref>{{Cite journal|date=2014-10-01|title=Foliated groupoids and infinitesimal ideal systems|journal=Indagationes Mathematicae|language=en|volume=25|issue=5|pages=1019–1053|doi=10.1016/j.indag.2014.07.009|issn=0019-3577|last1=Jotz Lean|first1=M.|last2=Ortiz|first2=C.|s2cid=121209093 |doi-access=free}}</ref>

=== Morphisms ===
A '''Lie algebroid morphism''' between two Lie algebroids <math>(A_1, [\cdot,\cdot]_{A_1}, \rho_1)</math> and <math>(A_2, [\cdot,\cdot]_{A_2}, \rho_2)</math> with the same base <math>M</math> is a vector bundle morphism <math>\phi: A_1 \to A_2</math> which is compatible with the Lie brackets, i.e. <math>\phi ([\alpha,\beta]_{A_1}) = [\phi(\alpha),\phi(\beta)]_{A_2}</math> for every <math>\alpha,\beta \in \Gamma(A_1)</math>, and with the anchors, i.e. <math>\rho_2 \circ \phi = \rho_1</math>.

A similar notion can be formulated for morphisms with different bases, but the compatibility with the Lie brackets becomes more involved.<ref>{{Cite book|last=Mackenzie|first=Kirill C. H.|url=https://www.cambridge.org/core/books/general-theory-of-lie-groupoids-and-lie-algebroids/DA70C6FAF52F88FB471F62DD68049608|title=General Theory of Lie Groupoids and Lie Algebroids|date=2005|publisher=Cambridge University Press|isbn=978-0-521-49928-6|series=London Mathematical Society Lecture Note Series|location=Cambridge|doi=10.1017/cbo9781107325883}}</ref> Equivalently, one can ask that the graph of <math>\phi: A_1 \to A_2</math> to be a subalgebroid of the direct product <math>A_1 \times A_2</math> (introduced below).<ref>Eckhard Meinrenken, [http://www.math.toronto.edu/mein/teaching/MAT1341_LieGroupoids/Groupoids.pdf Lie groupoids and Lie algebroids], Lecture notes, fall 2017</ref>

Lie algebroids together with their morphisms form a [[Category (mathematics)|category]].