Noncommutative geometry of the dihedral group D 6

TL;DR

This paper explores the noncommutative geometric structure of the dihedral group D6, including connections, curvature, and the Dirac operator, advancing the understanding of quantum geometric properties of finite groups.

Contribution

It explicitly constructs the torsion-free spin connection, computes curvature, and derives the Dirac operator eigenvalues for D6, providing new insights into quantum geometry of finite groups.

Findings

Explicit torsion-free spin connection for D6

Computed Riemann curvature and Ricci tensor

Derived eigenvalues and modes of the Dirac operator

Abstract

We study the noncommutative geometry of the dihedral group D 6 using the tools of quantum group theory. We explicit the torsion free regular spin connection and the corresponding 'Levi-Civita' connection. Next, we nd the Riemann curvature and its Ricci tensor. The main result is the Dirac operator of a representation of the group which we nd the eigenvalues and the eigenmodes.