Noncommutative Riemannian geometry of the alternating group [formula omitted

Abstract

We study the noncommutative Riemannian geometry of the alternating group A 4=( Z 2× Z 2)⋊ Z 3 using the recent formulation for finite groups. We find a unique ‘Levi-Civita’ connection for the invariant metric, and find that it has Ricci flat but nonzero Riemann curvature. We show that it is the unique Ricci flat connection on A 4 with the standard framing (we solve the vacuum Einstein’s equation). We also propose a natural Dirac operator for the associated spin connection and solve the Dirac equation. Some of our results hold for any finite group equipped with a cyclic conjugacy class of four elements. In this case the exterior algebra Ω( A 4) has dimensions 1:4:8:11:12:12:11:8:4:1 with top-form nine-dimensional. We also find the noncommutative cohomology H 1( A 4)= C .