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).
Reviewer: Martin Schlichenmaier (Mannheim)
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 |
Keywords:
Lie algebroid; exterior derivative; Poisson structure; vector bundle; Lie bialgebra; Leibniz structureReferences:
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