Manchester Geometry Seminar 2006/2007

12 October 2006. Room M12, Mathematics and Social Sciences building, Sackville Street. 4pm

Drinfeld's Classical Double and Mackenzie Theory

Theodore Voronov (University of Manchester)

Drinfeld's classical double (a counterpart of his "quantum double" for Hopf algebras) is a crucial object in the theory of Lie bialgebras, which are infinitesimal objects for Poisson-Lie groups and can be ultimately regarded as classical limits of quantum groups. Its analog for Lie bialgebroids (a notion discovered by Mackenzie and Xu) has remained a puzzle for a long time, although various constructions were suggested, seemingly non-equivalent. As it has turned out, the solution is in the combination of an approach based on supermanifolds with the beautiful Mackenzie theory of higher (e.g., double) structures "in the sense of Ehresmann",-- in particular, double Lie algebroids.

By combining the two, previously distant, approaches, the speaker has managed to simply the theory of doubles substantially and arrive at a transparent geometric picture. The talk is based on the paper "Mackenzie theory and Q-manifolds", arXiv:math.DG/0608111, by the speaker, and on a joint work in progress with Kirill Mackenzie, under the title "The double of a Lie bialgebroid is a double Lie bialgebroid". Our slogan is: the double of an n-fold Lie bialgebroid is an (n+1)-fold Lie bialgebroid.