Details (rather incomplete) of each week's talks:
The word `double' in the title refers to both the classical Drinfel'd double of a Lie bialgebra, and generalizations of it, and to abstractions of the iterated tangent bundle of a manifold. Structures of this type play an important role in Poisson geometry, and for the study of second-order constructions in differential geometry generally.
Mackenzie has defined a concept of double Lie algebroid ((math.DG/0611799) which abstracts the relations between two Lie algebroid structures, as in the cotangent double of a Lie bialgebroid, and the infinitesimal structure of a double Lie groupoid. This concept, given entirely in terms of standard differential geometry, is quite complicated.
Voronov (math.DG/0608111) reformulated Mackenzie's concept in super terms. Assuming one knows the super language, this approach is extremely efficient and seems likely to extend readily to the general multiple case.
This work is part of the theory of Lie groupoids and Lie algebroids. Lie algebroids encompass many fundamental constructions in differential geometry and mathematical physics - most first-order invariants of geometric structures are Lie algebroids. Lie groupoids provide global forms of Lie algebroids: as one example, a symplectic Lie groupoid integrating the cotangent Lie algebroid of a Poisson manifold provides a full realization of the Poisson manifold.
February 21, 2009