Editing Lie algebroid (section)

=== 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>