Noncommutative Riemannian geometry of the alternating group A(4)

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

\Omega(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

.

Note:

28 pages Latex no figures Subj-class: Quantum Algebra; Algebraic Geometry; Group Theory

References(8)

Figures(0)

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