TANGENT LIE ALGEBROIDS

This paper shows that a Lie algebroid structure on a smooth vector bundle A --> pi Q gives rise to a Lie algebroid structure on the bundle T A --> Tpi T Q, called the tangent Lie algebroid. The analysis uses global arguments. A Lie algebroid A is equivalent to a certain Poisson structure on A*, and the tangent bundle of any Poisson manifold has a tangent Poisson structure. The tangent Poisson structure on T A* is then dualized to produce the tangent Lie algebroid structure on T A. Local calculations are used, and formulae for local brackets are given.