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Noncommutative Riemannian geometry of the alternating group A4

✍ Scribed by F. Ngakeu; S. Majid; D. Lambert

Publisher: Elsevier Science
Year: 2002
Tongue: English
Weight: 167 KB
Volume: 42
Category: Article
ISSN: 0393-0440
DOI: 10.1016/s0393-0440(01)00089-4

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✦ Synopsis

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

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