On Contact and Symplectic Lie Algebroids

In this paper, we will study compatible triples on Lie algebroids. Using a suitable decomposition for a Lie algebroid, we construct an integrable generalized distribution on the base manifold. As a result, the symplectic form on the Lie algebroid induces a symplectic form on the leaves of the distribution. Hence, the induced Poisson structure on the base manifold can be represented by means of the induced Poisson structures on the integral submanifolds. Moreover, for any compatible triple with an invariant metric and an admissible almost complex structure, we show that the bracket annihilates on the kernel of the anchor map.