Noncommutative Riemannian geometry of the alternating group \({\mathcal A}_{4}\) (Q1603281)

The authors work out in detail an example illustrating a constructive approach to noncommutative Riemannian geometry based on quantum groups and developed in [\textit{S. Majid}, J. Geom. Phys. 30, 113-146 (1999; Zbl 0940.58004) and Commun. Math. Phys. 225, 131-170 (2002; Zbl 0999.58004)]. Within this formalism one can equip finite sets with the Riemannian manifold structure. It is shown that the alternating group \({\mathcal A}_4\) has a Riemannian structure in this sense. A natural invariant metric with unique Levi-Civita connection which is Ricci flat but has non-zero Riemannian curvature is found. By this means vacuum Einstein's equations on \({\mathcal A}_4\) are solved. A Dirac operator for associated spin connection is proposed and the Dirac equation is solved. The exterior algebra of \({\mathcal A}_4\) is also studied, and the corresponding first (noncommutative de Rham) cohomology group of \({\mathcal A}_4\) is shown to be equal to \(\mathbb C\).