Algebra of Path Integrals on Digraphs

TL;DR

This paper extends the concept of iterated integrals to directed graphs, establishing algebraic structures like Hopf algebras and linking them to fundamental groups, thus broadening the mathematical framework for analyzing paths on digraphs.

Contribution

It introduces a novel algebraic framework for iterated integrals on digraphs, including the construction of associated Hopf algebras and their relation to fundamental groups.

Findings

Development of iterated path and loop algebras on digraphs

Construction of a Hopf algebra structure with involutive antipode

Establishment of a homotopy invariant subalgebra linked to the fundamental group

Abstract

In this paper, we extend the iterated integrals from smooth manifolds to digraphs and develop the associated algebraic and geometric structures. Iterated integrals on a digraph naturally give rise to the iterated path algebra and the iterated loop algebra, both defined as quotient algebras of a shuffle algebra, with the latter carrying a canonical Hopf algebra structure. We construct a non-degenerate pairing between elementarily equivalent classes of loops on a digraph and the iterated loop algebra. By restricting to iterated integrals that are invariant under $C_\partial$-homotopy, a distinguished subalgebra is obtained which, under this pairing, corresponds to the group algebra of the fundamental group. We further show that this subalgebra is a homotopy invariant and forms a Hopf algebra with involutive antipode.