### University of Manchester

### Informal Research Seminar on Geometry and
Mathematical Physics

### Special programme:

## Double and Multiple
Structures in Standard and Super Geometry

Details (rather incomplete) of each week's talks:

#### Description of the programme:

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.

The seminar is partially supported by
MIMS, an institute of the
School of Mathematics at the University of Manchester.

February 21, 2009