Manchester Geometry Seminar 2000/2001

Classical Drinfeld's double is a construction that makes a new Lie bialgebra, with nice properties, of a given Lie bialgebra. (This is a "quasi-classical counterpart" of quantum double construction for Hopf algebras. A Lie bialgebra is a Lie algebra such that the dual space is also a Lie algebra, with a natural compatibility condition.)

I will define graded manifolds as supermanifolds with a Z-grading ("weight") in the structure sheaf. The relation will be explained between Lie (bi)algebras and such structures on supermanifolds as homological fields, Poisson and Schouten (Gerstenhaber) brackets. I will give a construction of doubles for graded QS- and graded QP-manifolds. This is a far-going generalization of Drinfeld's bialgebras and their doubles, and at the same time provides a very natural simple framework for the original Drinfeld's construction, in terms of super Poisson geometry. In particular, this is closely related with the analogue of Drinfeld's double for Lie bialgebroids recently suggested by Roytenberg and Vaintrob. Lie bialgebroids as "relative" version of Lie bialgebras, over some base, were defined by Kirill Mackenzie and Ping Xu. They are infinitesimal version of Poisson groupoids and quantum groupoids.