Graded manifolds and Drinfeld doubles for Lie bialgebroids

Abstract

We define graded manifolds as a version of supermanifolds endowed with an extra Z-grading in the structure sheaf, called weight (not linked with parity). Examples are ordinary supermanifolds, vector bundles, double vector bundles (in particular, iterated constructions like TTM), etc. I give a construction of doubles for graded QS- and graded QP-manifolds (graded manifolds endowed with a homological vector field and a Schouten/Poisson bracket). Relation is explained with Drinfeld's Lie bialgebras and their doubles. Graded QS-manifolds can be considered, roughly, as generalized Lie bialgebroids. The double for them is closely related with the analog of Drinfeld's double for Lie bialgebroids recently suggested by Roytenberg. Lie bialgebroids as a generalization of Lie bialgebras, over some base manifold, were defined by Mackenzie and P. Xu. Graded QP-manifolds give an version for all this, in particular, they contain odd analogs for Lie bialgebras, Manin triples, and Drinfeld's double.