University of Toronto's Symplectic Geometry Seminar



September 14, 2009, 2:10 pm
Bahen 6183



Alejandro Cabrera

University of Toronto

Lie theory and supergeometry




Abstract:

In this talk we will study multiplicative differential forms and multivectors on Lie groupoids. These are forms and multivectors that satisfy a certain compatibility condition with groupoid multiplication. Examples of such things include Poisson-Lie groups and symplectic groupoids. Our main goal is to characterize these multiplicative objects in terms of infinitesimal data, this is, in terms of the corresponding "infinitesimally multiplicative" (IM) stuctures induced on the underlying Lie algebroid. We thus explain 1:1 type of results for source-simply-connected groupoids between the infinitesimal and the global structures under study. We also explain how the "Lie functor" taking multiplicative structures on the groupoid to IM-structures on its algebroid commutes with natural operations such as de Rham differential and Schoutens bracket. Finally, we shall point out how one can use certain "supergeometric intuition procedure" to get to all the above results (in all of their details) by extrapolation of much simpler ones.