Isotopy Quotients of Hopf Algebroids and the Fundamental Groupoid of Digraphs

TL;DR

This paper establishes a Riemann-Hilbert correspondence for digraphs by linking the fundamental groupoid's algebra to isotopy quotients of Hopf algebroids, advancing the understanding of noncommutative geometric structures.

Contribution

It introduces the concepts of Hopf ideals and isotopy quotients for Hopf algebroids and proves a new correspondence connecting groupoid algebras to these quotients.

Findings

Groupoid algebra of a digraph is isomorphic to an isotopy quotient of a Hopf algebroid.

Construction of Hopf algebroids from noncommutative calculi with surjectivity.

Establishment of a Riemann-Hilbert correspondence for digraphs.

Abstract

We build on our construction of Hopf algebroids from noncommutative calculi under the further assumption of surjectivity for the calculus. We also introduce the notions of Hopf ideals and isotopy quotients for arbitrary Hopf algebroids. Using these ingredients, we prove a Riemann-Hilbert correspondence for digraphs, by showing that the groupoid algebra of the fundamental groupoid of a digraph is isomorphic to the isotopy quotient of the Hopf algberoid corresponding to flat connections over the digraph.