Lie Algebroid

Lie Algebroid

In mathematics, Lie algebroids serve the same role in the theory of Lie groupoids that Lie algebras serve in the theory of Lie groups: reducing global problems to infinitesimal ones. Just as a Lie groupoid can be thought of as a "Lie group with many objects", a Lie algebroid is like a "Lie algebra with many objects".

More precisely, a Lie algebroid is a triple consisting of a vector bundle over a manifold, together with a Lie bracket on its module of sections and a morphism of vector bundles called the anchor. Here is the tangent bundle of . The anchor and the bracket are to satisfy the Leibniz rule:

where and is the derivative of along the vector field . It follows that

for all .