Generalised noncommutative geometry on finite groups and Hopf quivers

TL;DR

This paper develops a framework for noncommutative differential geometry on finite sets using quivers, connecting it to Hopf algebra structures and quantum principal bundles, with explicit examples and duality insights.

Contribution

It introduces a novel approach to finite noncommutative geometry via quivers, linking differential calculus to Hopf quiver data and exploring duality and quantum bundle structures.

Findings

Describes differential calculus on finite sets using quivers.

Establishes a duality between geometry on function and group algebras.

Provides explicit examples of quantum metrics on quivers.

Abstract

We explore the differential geometry of finite sets where the differential structure is given by a quiver rather than as more usual by a graph. In the finite group case we show that the data for such a differential calculus is described by certain Hopf quiver data as familiar in the context of path algebras. We explore a duality between geometry on the function algebra vs geometry on the group algebra, i.e. on the dual Hopf algebra, illustrated by the noncommutative Riemannian geometry of the group algebra of $S_3$. We show how quiver geometries arise naturally in the context of quantum principal bundles. We provide a formulation of bimodule Riemannian geometry for quantum metrics on a quiver, with a fully worked example on 2 points; in the quiver case, metric data assigns matrices not real numbers to the edges of a graph. The paper builds on the general theory in our previous work.