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 .
Read more about Lie Algebroid: Examples, Lie Algebroid Associated To A Lie Groupoid
Famous quotes containing the word lie:
“Mind not the old man beseeching the young man;
Let not the child’s voice be heard, nor the mother’s entreaties;
Make even the trestles to shake the dead, where they lie awaiting
the hearses,
So strong you thump, O terrible drums—so loud you bugles blow.”
—Walt Whitman (1819–1892)