Tangent Lie algebroids

Abstract

This paper shows that a Lie algebroid structure on a smooth vector bundle A to ( pi over)Q gives rise to a Lie algebroid structure on the bundle TA to (T pi over)TQ, 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 TA* is then dualized to produce the tangent Lie algebroid structure on TA. Local calculations are used, and formulae for local brackets are given.