Tangent Lie algebroids. (English) Zbl 0843.58044

Courant, Ted

J. Phys. A, Math. Gen. 27, No. 13, 4527-4536 (1994).

Summary: 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 \(TA@> T\pi>> TQ\), 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.

MSC:

37J99

Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems

Keywords:

Lie algebroid; Poisson structure