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

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

2006-11-04

arxiv.org/abs/math/0611100v2

Commun. Math. Phys. 279 (2008) 77--116

Mathematics

Quantum Algebra

40 pages, no figures, Latex. v2: Title changed. Sect. 9 on real structure completely rewritten and results strengthened. Addit

Scientific paper

10.1007/s00220-008-0420-x

Equivariance under the action of Uq(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 this sphere with a Dirac operator which is isospectral to the canonical one on the round undeformed four-sphere and which gives metric dimension four for the noncommutative geometry. 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.

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