# Generalised noncommutative geometry on finite groups and Hopf quivers

### Shahn Majid

Queen Mary University of London, UK### Wen-Qing Tao

Huazhong University of Science and Technology, Wuhan, China

## 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 [19].

## Cite this article

Shahn Majid, Wen-Qing Tao, Generalised noncommutative geometry on finite groups and Hopf quivers. J. Noncommut. Geom. 13 (2019), no. 3, pp. 1055–1116

DOI 10.4171/JNCG/345