Lie algebroid

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In mathematics, a Lie algebroid is a triple (E,[,],ρ) consisting of a vector bundle E over a manifold M, together with a Lie bracket [,] on its module of sections Γ(E) and a morphism of vector bundles called the anchor. Here TM is the tangent bundle of M. The anchor and the bracket are to satisfy the conditions:

1) ρ([X,Y]) = [ρ(X),ρ(Y)]

2)

where and ρ(X)f is the derivative of f along the vector field ρ(X)

Examples:

1) To every Lie groupoid is associated a Lie algebroid analogous to the correspondence of a Lie algebra to a Lie group.

2) Given the action of a Lie Algebra g on a Manifold M, the set of g -invariant vector fields on M is a Lie Algebroid over the space of orbits of the action.