An Isaac Newton Institute Workshop

Trends in Noncommutative Geometry

NONLINEAR TRANSFORMATIONS IN LAGRANGIANS AND A GRAPHICAL CALCULUS

Author: Pierre CARTIER (IHES Bures )

Abstract

We describe first a general graphical method to represent nonlinear transformations , and apply it to the case of the invariant transformations of tensors of a certain kind . We interpret then the Lie algebra as well as the Hopf algebra of functions of such a group of nonlinear transformations . Finally we show how to connect these constructions with the Hopf algebra introduced by Connes and Kreimer in Renormalization Theory . As a bonus , we settle a vexing question of normalization connected with the number of symmetries of a Feynman diagram , and we recover a theorem of Connes and Kreimer about the renormalization group .