Algebroids – General Differential Calculi on Vector Bundles

A notion of an algebroid-a generalization of a Lie algebroid structure on a vector bundle is introduced. We show that many objects of the differential calculus on a manifold M associated with the canonical Lie algebroid structure on TM can be obtained in the framework of a general algebroid. Also a compatibility condition which leads, in general, to a concept of a bialgebroid. 0 Introduction. The classical Cartan differential calculus on a manifold M , including the exterior derivative d, the Lie derivative £, etc., can be viewed as being associated with the canonical Lie algebroid structure on TM represented by the Lie bracket of vector fields. Lie algebroids have been introduced repeatedly into differential geometry since the early 1950s, and also into physics and algebra, under a wide variety of names. They have been also recognized as infinitesimal objects for Lie groupoids ([18]). We refer to [14] for basic definitions, examples, and an extensive list of publications in these directions. Being related to many areas of geometry, like connection theory, cohomology theory, invariants of foliations and pseudogroups, symplectic and Poisson geometry, etc., Lie alge-broids became recently an object of extensive studies. What we propose in this paper is to find out what are, in fact, the structures responsible for the presence of a version of the Cartan differential calculus on a vector bundle and how are they related. This leads to the notion of a general algebroid. It is well known that there exists a one-one correspondence between Lie algebroid structures on a vector bundle τ : E → M and linear Poisson structures on the dual vector bundle π: E * → M. This correspondence can be extended to much wider class of binary operations (brackets) on sections of τ on one side, and linear contravariant 2-tensor fields on E * on the other side. It is not necessary for these operations to be skew-symmetric or to satisfy the Jacobi identity. The vector bundle τ together with a bracket operation, or the equivalent contravariant 2-tensor field, will be called an algebroid. This terminology is justified by the fact that contravariant 2-tensor fields define certain binary operations on the space C ∞ (E *). The algebroids constructed in this way include all finite-dimensional algebras over real numbers (e.g. associative, Jordan, etc.) as particular examples. The base manifold M is in these cases a single point. Searching for structures which …