Algebra is Geometry is Algebra – Interactions Between Hopf Algebras, Infinite Dimensional Geometry and Application

Abstract

This chapter examines the interaction of algebra and geometry in the guise of Hopf algebras and certain associated character groups. The geometry mirrors the algebra in that equation becomes a Lie group anti-homomorphism. Furthermore, the geometric structure allows us to give intrinsic geometric meaning of certain constructions in numerical analysis, such as Lie derivatives and differential equations, on the groups. Different Hopf algebras and their characters are studied from the perspectives of numerical analysis, renormalization of quantum field theories, the theory of rough paths and control theory. The base is part of the structure of a combinatorial Hopf algebra and in applications consists of combinatorial objects like trees, graphs, words or permutations. In general, their category of combinatorial Hopf algebras and our category CombHopf are incomparable, as the notion of combinatorial Hopf algebras is incomparable. The chapter considers all the classical examples of combinatorial Hopf algebras contained in both categories of combinatorial Hopf algebras.