Anchored Vector Bundles and Algebroids

Inspired by recent works of Zang Liu, Alan Weinstein and Ping Xu, we introduce the notions of CC algebroids and non asymmetric Courant algebroids and study these structures. It is shown that CC algebroids of rank greater than 3 are the same as Courant algebroids up to a constant factor, though the definition of CC algebroids is much simpler than that of Courant algebroids,requiring only 2 axioms instead of 5. The situation is similar to that of Lie algebroids, where in the usual definition used by all of he experts there is a redundant axiom, e.g.[GG,KO1,KO2,MK,PL]. Non asymmetric Courant algebroids are shown to be nothing but (pseudo)clan bundles (in the sense of E.B. Vinberg-Katz) which arise in affine geometry of convex bounded domains. The study of CC algebroids and non asymmetric Courant algebroids involves the cohomology theory of Koszul-Vinberg algebras and their modules.