Tulczyjew triples and higher Poisson/Schouten structures on Lie algebroids

We show how to extend the construction of Tulczyjew triples to Lie algebroids via graded manifolds. We also provide a generalisation of triangular Lie bialgebroids as higher Poisson and Schouten structures on Lie algebroids.Comment: 28 pages. Completely rewritten and improved. Typos corrected. A version is to appear in Reports on Mathematical Physics, Vol.66, No. 2, 2010. Further minor typos correcte