Abstract:In this paper, we extend the iterated path integrals from smooth manifolds to digraphs and develop the associated algebraic and geometric structures. Iterated path 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 path 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.
From: Yunpeng Zi [view email]
[v1]
Mon, 2 Mar 2026 07:00:53 UTC (497 KB)
[v2]
Fri, 28 Aug 2026 06:26:57 UTC (168 KB)
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