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Concept
Lie algebroid
Résumé
In mathematics, a Lie algebroid is a vector bundle together with a Lie bracket on its space of sections and a vector bundle morphism , satisfying a Leibniz rule. A Lie algebroid can thus be thought of as a "many-object generalisation" of a Lie algebra.
Lie algebroids play a similar same role in the theory of Lie groupoids that Lie algebras play in the theory of Lie groups: reducing global problems to infinitesimal ones. Indeed, any Lie groupoid gives rise to a Lie algebroid, which is the vertical bundle of the source map restricted at the units. However, unlike Lie algebras, not every Lie algebroid arises from a Lie groupoid.
Lie algebroids were introduced in 1967 by Jean Pradines.
A Lie algebroid is a triple consisting of
a vector bundle over a manifold
a Lie bracket on its space of sections
a morphism of vector bundles , called the anchor, where is the tangent bundle of
such that the anchor and the bracket satisfy the following Leibniz rule:
where and is the derivative of along the vector field .
One often writes when the bracket and the anchor are clear from the context; some authors denote Lie algebroids by , suggesting a "limit" of a Lie groupoids when the arrows denoting source and target become "infinitesimally close".
It follows from the definition that
for every , the kernel is a Lie algebra, called the isotropy Lie algebra at
the kernel is a (not necessarily locally trivial) bundle of Lie algebras, called the isotropy Lie algebra bundle
the image is a singular distribution which is integrable, i.e. its admits maximal immersed submanifolds , called the orbits, satisfying for every . Equivalently, orbits can be explicitly described as the sets of points which are joined by A-paths, i.e. pairs of paths in and in such that and
the anchor map descends to a map between sections which is a Lie algebra morphism, i.e.
for all .
The property that induces a Lie algebra morphism was taken as an axiom in the original definition of Lie algebroid. Such redundancy, despite being known from an algebraic point of view already before Pradine's definition, was noticed only much later.
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