An extension of the Baker-Campbell-Hausdorff formula to higher groups
by Professor Ezra Getzler
Abstract: In modern language, Lie showed, as an application of the theory of ordinary differential equations, that there is a correspondence between nilpotent Lie algebras and simply connected nilpotent Lie groups. Quillen found a natural extension of this result: there is an equivalence between nilpotent differential graded Lie algebras and topological spaces with nilpotent rational homotopy. This theorem has many applications, for example, understanding the deformation theory of geometric structures such as Poisson manifolds and Poisson-Nijenhuis manifolds (the latter arise in the theory of completely integrable systems), and the study of higher gauge fields in supersymmetric fields theories (whose potential is a higher-degree differential form, generalizing Maxwell’s theory).
In this talk, I will describe an extension of the Baker-Campbell-Hausdorff formula to Quillen’s theorem, which gives equations for the correspondence between nilpotent differential graded Lie algebras and nilpotent spaces.