Courant Research Centres ´´Higher Order Structures in Mathematics´´

Research Group: Differential geometry (Chengchang Zhu.)

We are interested in Lie theory of general symmetries, for example that of Lie algebroids (which can be regarded as degree 1 super-manifolds with a degree 1 vector field Q, such that [Q,Q] = 0)

We start by a simple example: Given a circle x²+y²=1, the tangent vector at point(x,y) is v(y)= (x,y). On the other hand, if we are given a series of vectors like these, we can follow these vectors infinitesimally and get back to the circle as the global object. This allows us to encode the global symmetry by local data and vice-versa. Sophus Lie and various great mathematicians (many of whom were in Göttingen) summarized this as the theory of Lie algebras and Lie groups, which is one of the greatest achievements in 19th century mathematics.

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However there are other sorts of symmetries not included in Lie's classical theory. We take as global object the set of pairs of points (p,q) in a space, for example the sphere what is the infinitesimal object? As two points get closer and closer, they become a tangent vector at the limiting point. Hence the infinitesimal object is the set of all the tangent spaces at all points on This sort of symmetry correspondence is summarized in the theory of Lie algebroids and Lie groupoids. However, unlike the classical Lie theory, in this case, the infinitesimal data (called by mathematicians a Lie algebroid) might not always close up to a global object in a usual sense. It turns out the global object has to be a sort of stacky Lie groupoid to fullfill the 1-1 correspondence.

liegrupo.jpg

This launched various efforts of modern mathematicians, and has relation to many other directions, such as Poisson sigma models, super manifolds, the theory of differentiable stacks and Lie groupoids, higher groupoids, symmetries in generalized complex geometry, integration problems on Poisson manifolds, Jacobi manifolds, and Dirac manifolds.