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Algebroids – general differential calculi on vector bundles. (English) Zbl 0954.17014
In this article the notion of an algebroid is introduced. It is a generalization of a Lie algebroid structure on a vector bundle. A vector bundle over a manifold together with a bracket operation, or the equivalent contravariant 2-tensor field, is called an algebroid. It is shown that many objects of the differential calculus on a manifold associated with the canonical Lie algebroid structure of the tangent bundle has an extension to the algebroid setting. A compatibility condition is studied which leads to the concept of a bialgebroid. The constructions are done with the help of Leibniz structures on manifolds. Note that certain generalizations of Lie algebroids have been around earlier (e.g. Jacobian quasi-bialgebras, Loday algebras, pre-Lie algebroids, as introduced by Kosmann-Schwarzbach, Magri, Loday, and others).

##### MSC:
 17B66 Lie algebras of vector fields and related (super) algebras 58H05 Pseudogroups and differentiable groupoids 53D17 Poisson manifolds; Poisson groupoids and algebroids 17B63 Poisson algebras
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