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Algebra of Path Integrals on Digraphs

Shing-Tung Yau, Mengmeng Zhang, Yunpeng Zi

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.

Algebra of Path Integrals on Digraphs

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 -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.
Paper Structure (14 sections, 35 theorems, 116 equations, 2 figures, 1 table)

This paper contains 14 sections, 35 theorems, 116 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Differential Forms on Digraphs
  4. Homotopy Theory on Digraphs
  5. Iterated Integrals on the Digraphs
  6. Basic Properties of Iterated Integrals
  7. Elementary Equivalence
  8. Bilinear Form and Adjointness
  9. Algebras on Digraphs
  10. Shuffle, Loop and Iterated Algebras
  11. Filtration
  12. Homotopy invariant algebras $\pi^1$
  13. Hopf Algebra Structure
  14. Change of the Base Point

Key Result

Theorem 1.1

Two path maps $\alpha$ and $\beta$ are elementary equivalent if and only if $\int_{\alpha}\omega_1\cdots\omega_r=\int_{\beta}\omega_1\cdots\omega_r$ for any 1-forms $\omega_1,\cdots,\omega_r$, with $r\geq 1$.

Figures (2)

  • Figure 1: Three contractible digraphs.
  • Figure 2: Visualization for $r=2$. For the point $(t_1,t_2)=(3,4)$, $\tau(3,4)=1$ and for the point $(t_1,t_2)=(5,5)$, $\tau(5,5)=2$

Theorems & Definitions (75)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 2.1
  • Theorem 2.2
  • Definition 3.1
  • Lemma 3.2: Equivalent Definition of Volume number
  • Remark 3.3
  • Definition 3.4
  • Definition 3.5
  • Proposition 3.6
  • ...and 65 more