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Noncommutative Riemannian Geometry of the Alternating Group A_4

F. Ngakeu·S. Majid·D. Lambert
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Abstract

We study the noncommutative Riemannian geometry of the alternating group $A_4=(Z_2 \times Z_2)\cross Z_3$ using a 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 4 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 9-dimensional. We also find the noncommutative cohomology $H^1(A_4)=C$.

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BibTeXRIS

F. Ngakeu, S. Majid, D. Lambert. 2001-07-30. Noncommutative Riemannian Geometry of the Alternating Group A_4. https://doi.org/10.1016/s0393-0440(01)00089-4

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