Lie algebroids were introduced in 1967 by Jean Pradines.
consisting of such that the anchor and the bracket satisfy the following Leibniz rule: where
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".
[1] Such redundancy, despite being known from an algebraic point of view already before Pradine's definition,[3] was noticed only much later.
For this reason, the more flexible notion of infinitesimal ideal system has been introduced.
A similar notion can be formulated for morphisms with different bases, but the compatibility with the Lie brackets becomes more involved.
A Lie algebroid is called totally intransitive if the anchor map
This actually exhaust completely the list of totally intransitive Lie algebroids: indeed, if
is totally intransitive, it must coincide with its isotropy Lie algebra bundle.
A Lie algebroid is called transitive if the anchor map
For instance: In analogy to Atiyah algebroids, an arbitrary transitive Lie algebroid is also called abstract Atiyah sequence, and its isotropy algebra bundle
A Lie algebroid is called regular if the anchor map
must be trivial, therefore both conditions are empty, and we recover the standard notion of action of a Lie algebra on a manifold.
denotes the Lie derivative with respect to the vector field
, we recover the standard notion of connection on a vector bundle, as well as those of curvature and flatness.
has a natural structure of semiring, with direct sums and tensor products of vector bundles.
defined as follows: Of course, a symmetric construction arises when swapping the role of the source and the target maps, and replacing right- with left-invariant vector fields; an isomorphism between the two resulting Lie algebroids will be given by the differential of the inverse map
This allows one to defined the analogue of the exponential map for Lie groups as
Then Let us describe the Lie algebroid associated to the pair groupoid
is just the Lie bracket of tangent vector fields and the anchor map is just the identity.
is given by the differential of the target map, there are two cases for the isotropy Lie algebras, corresponding to the fibers of
is an integrable Lie algebroid, then there exists a unique (up to isomorphism)
The analogue of the classical Lie II theorem states that:[13] if
-simply connected, then there exists a unique morphism of Lie groupoids
Pradines claimed that such a statement hold,[14] and the first explicit example of non-integrable Lie algebroids, coming for instance from foliation theory, appeared only several years later.
[15] Despite several partial results, including a complete solution in the transitive case,[16] the general obstructions for an arbitrary Lie algebroid to be integrable have been discovered only in 2003 by Crainic and Fernandes.
-simply connected topological groupoid, with the multiplication induced by the concatenation of paths.
This approach led to the introduction of objects called monodromy groups, associated to any Lie algebroid, and to the following fundamental result:[17] A Lie algebroid is integrable if and only if its monodromy groups are uniformly discrete.Such statement simplifies in the transitive case:A transitive Lie algebroid is integrable if and only if its monodromy groups are discrete.The results above show also that every Lie algebroid admits an integration to a local Lie groupoid (roughly speaking, a Lie groupoid where the multiplication is defined only in a neighbourhood around the identity elements).
is transitive, it is integrable if and only if it is the Atyah algebroid of some principal bundle; a careful analysis shows that this happens if and only if the subgroup