Lie theory" refers to a functorial correspondence between

finite-dimensional Lie algebras and 1-connected Lie groups, elucidated

by Sophus Lie and summarized in his three theorems, Lie I, II and III.

The first hint that there might be more to the story came from van Est

who noticed (and proved) that integrating higher Lie algebra cocycles

required higher connectivity assumptions on the Lie group. Furthermore,

attempts to build a Lie theory for Lie algebroids (and

infinite-dimensional Lie algebras) generally run into obstructions of a

homotopy-theoretic nature, leading to higher structures -- Lie 2-groups

and groupoids. At the same time, it was discovered that many geometric

structures arising in theoretical physics exhibit a hierarchy of

higher-order symmetries (symmetries between symmetries, and so on) which

behave a lot like homotopy types in algebraic topology. The goal of

higher Lie theory is to construct a higher-categorical version of Lie's

correspondence for "spaces with infinitesimal higher symmetries"

(modelled by differential graded manifolds) on the one hand, and "spaces

with global higher symmetries" (modelled by simplicial manifolds) on the

other. In this talk we will explain the problem and the main concepts

and constructions involved, and survey the progress so far.

© MPI f. Mathematik, Bonn | Impressum & Datenschutz |