Anchored vector bundles and Lie algebroids (Q2782564)

The authors define an anchored vector bundle \(\theta\) to be a vector bundle over a manifold \(M\) endowed with a homomorphism (called the anchor) to the tangent bundle \(TM\). Given such a structure there are straightforward notions of a Lie bracket on \(\theta\) and of connections on vector bundles for which the tangent bundle is replaced by \(\theta\). For an anchored vector bundle endowed with an analog of the Lie bracket and a connection in the above sense, the authors next give a rather involved iterative construction of derived algebroids, and study functorial properties of this construction. In particular, they deduce that the union of all derived algebroids (which has infinite dimensional fibers) is independent of the choice of the bracket and connection. Next, they construct a quotient of this, which is a generalized Lie algebroid. Finally relations to foliations and generalizations of differential forms are discussed. Unfortunately, some of the proofs in the paper seem rather vague to me, and the authors neither worked out examples nor discussed possible applications.NEWLINENEWLINEFor the entire collection see [Zbl 0980.00030].