Graded generalized geometry

Generalized geometry finds many applications in the mathematical description of some aspects string theory. In a nutshell, it explores various structures on generalized tangent bundle associated to given manifold. particular, several integrability conditions can be formulated terms canonical Dorfman bracket, an example Courant algebroid. On other hand, smooth manifolds involve functions $\mathbb{Z}$-graded variables which do not necessarily commute. This leads theory graded manifolds. It is only natural combine two theories by exploring After recalling elementary geometry, algebroids vector bundles are introduced. We show that there bracket Graded analogues Dirac and complex explored. introduce differential viewed as generalization Q-manifolds. A definition examples Lie bialgebroids given.