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Generalised noncommutative geometry on finite groups and Hopf quivers

Journal Contribution - Journal Article

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].
Journal: Journal of Noncommutative Geometry
ISSN: 1661-6952
Issue: 3
Volume: 13
Pages: 1055 - 1116
Publication year:2019
Keywords:Hopf algebra, nonsurjective calculus, quiver, duality, finite group, bimodule connection
BOF-keylabel:yes
IOF-keylabel:yes
BOF-publication weight:1
CSS-citation score:1
Authors:International
Authors from:Higher Education
Accessibility:Closed