, Volume 279, Issue 1, pp 77-116
Date: 08 Feb 2008

The Isospectral Dirac Operator on the 4-dimensional Orthogonal Quantum Sphere

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Abstract

Equivariance under the action of U q (so(5)) is used to compute the left regular and (chiral) spinorial representations of the algebra of the orthogonal quantum 4-sphere \(S^4_q\) . These representations are the constituents of a spectral triple on \(S^4_q\) with a Dirac operator which is isospectral to the canonical one on the round sphere S 4 and which then gives 4+-summability. Non-triviality of the geometry is proved by pairing the associated Fredholm module with an ‘instanton’ projection. We also introduce a real structure which satisfies all required properties modulo smoothing operators.

Communicated by A. Connes