From Iterated Integrals and Chronological Calculus to Hopf and Rota–Baxter Algebras

Abstract

This chapter begins by examining the Hopf algebraic interpretation of classical iterated integral calculus and its rooting in the geometry of simplices. It includes the case of tree-shaped iterated integrals, which are relevant in many situations. The chapter examines algebraic structures underlying calculus with iterated integrals and show how they lead naturally to the notions of descent (Hopf) algebra as well as permutation Hopf algebra. It shows how to handle algebraically products of iterated integrals parametrized by permutations. The chapter introduces chronological calculus and the associated pre-Lie structures, together with the construction of their enveloping algebras. It outlines the main algebraic aspects of the contemporary theory of Rota–Baxter algebras. The interest in Rota–Baxter algebras results from the possibility to describe interesting and useful universal combinatorial identities that hold in a wide variety of contexts. The chapter argues that the construction of the descent algebra was motivated by the combinatorial formulas for products of iterated integrals.