Algebra of Path Integrals on Digraphs

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.