# 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)

`theodore.voronov@manchester.ac.uk`

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*.