This mini course discusses on the one hand recent results on homotopy of Lie algebroids versus twisted Lie algebroid cohomology, and on the other hand the notion of ideal in Lie algebroids, as well as obstructions to these structures. The plan of the lectures is the following:
The first lecture discusses Lie algebroids and their representations up to homotopy, twisted Lie algebroid cohomology and its homotopy invariance. This first lecture is based on joint work with Rosa Marchesini.
The second lecture introduces Pontryagin characters of graded vector bundles (of finite rank) and shows a simple obstruction result to a representation up to homotopy on a given graded vector bundle. The third lecture introduces the notion of ideal in a Lie algebroid, as structures allowing geometric reduction of Lie algebroids, and as subrepresentations up to homotopy of the adjoint representations. Then obstructions to ideal in Lie algebroids are discussed. The third lecture is partially based on joint work with Thiago Drummond and Cristian Ortiz.