Editing Lie algebroid (section)

=== Examples from differential geometry ===
* Given a [[foliation]] <math>\mathcal{F}</math> on <math>M</math>, its '''foliation algebroid''' is the associated involutive subbundle <math>\mathcal{F} \subseteq TM</math>, with brackets and anchor induced from the tangent Lie algebroid.
* Given the action of a Lie algebra ''<math>\mathfrak{g}</math>'' on a manifold <math>M</math>, its '''action algebroid''' is the trivial vector bundle <math>\mathfrak{g} \times M \to M</math>, with anchor given by the Lie algebra action and brackets uniquely determined by the bracket of <math>\mathfrak{g}</math> on constant sections <math>M \to \mathfrak{g}</math> and by the Leibniz identity.
*Given a [[principal bundle|principal ''G''-bundle]] ''<math>P</math>'' over a manifold ''<math>M</math>'', its '''[[Atiyah algebroid]]''' is the Lie algebroid <math>A = TP/G</math> fitting in the following [[short exact sequence]]:
*:<math> 0 \to \ker(\rho) \to TP/G\xrightarrow{\rho} TM \to 0.</math>
: The space of sections of the Atiyah algebroid is the Lie algebra of ''<math>G</math>''-invariant vector fields on ''<math>P</math>'', its isotropy Lie algebra bundle is isomorphic to the [[Adjoint bundle|adjoint vector bundle]] <math>P\times_G \mathfrak g</math>, and the right splittings of the sequence above are [[principal connection]]s on ''<math>P</math>.''
*Given a vector bundle <math>E \to M</math>, its '''general linear algebroid''', denoted by ''<math>\mathfrak{gl}(E)</math>'' or ''<math>\mathrm{Der}(E)</math>'', is the vector bundle whose sections are derivations of <math>E</math>, i.e. first-order [[differential operator]]s <math>\Gamma(E) \to \Gamma(E)</math> admitting a vector field <math>\rho(D) \in \mathfrak{X}(M)</math> such that <math>D(f \sigma) = f D(\sigma) + \rho(D)(f) \sigma</math> for every <math>f \in \mathcal{C}^{\infty}(M), \sigma \in \Gamma(E)</math>. The anchor is simply the assignment <math>D \mapsto \rho(D)</math> and the Lie bracket is given by the commutator of differential operators.
*Given a [[Poisson manifold]] ''<math>(M,\pi)</math>'', its '''cotangent algebroid''' is the cotangent vector bundle ''<math>A = T^*M</math>'', with Lie bracket ''<math>[\alpha,\beta]:= \mathcal{L}_{\pi^\sharp (\alpha)} (\beta) - \mathcal{L}_{\pi^\sharp (\beta)} (\alpha) - d \pi( \alpha, \beta)</math>'' and anchor map ''<math>\pi^\sharp: T^*M \to TM, \alpha \mapsto \pi(\alpha,\cdot)</math>''.
*Given a closed 2-form <math>\omega \in \Omega^2(M)</math>, the vector bundle ''<math>A_\omega := TM \times \mathbb{R} \to M</math>'' is a Lie algebroid with anchor the projection on the first component and Lie bracket''<math display="block">[(X,f), (Y,g)]:= \Big( [X,Y], \mathcal{L}_X(g) - \mathcal{L}_Y(f) - \omega(X,Y) \Big).</math>''Actually, the bracket above can be defined for any 2-form <math>\omega</math>, but <math>A_\omega</math> is a Lie algebroid if and only if <math>\omega</math> is closed.