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

Abstract

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.

40 pages, no figures, Latex. v2: Title changed. Sect. 9 on real structure completely rewritten and results strengthened. Additional minor changes throughout the paper

References in corpus (5)

• Gravity coupled with matter and foundation of non-commutative geometry
• On the eigenfunctions of the Dirac operator on spheres and real hyperbolic spaces
• The Dirac operator on SU_q(2)
• Local Index Formula on the Equatorial Podles Sphere
• Fredholm Modules for Quantum Euclidean Spheres

Cited by in corpus (10)

• Dirac Operators on Quantum Projective Spaces
• Anti-selfdual Connections on the Quantum Projective Plane: Monopoles
• The Noncommutative Geometry of the Quantum Projective Plane
• Quantum Isometries of the finite noncommutative geometry of the Standard Model
• The 3D Spin Geometry of the Quantum Two-Sphere
• Spectral action on SU_q(2)
• Noncommutative Geometry and Quantum Group Symmetries
• Geometry of the quantum projective plane
• Dirac operator on the quantum fuzzy four-sphere $S_{qF}^4$
• A Leray spectral sequence for noncommutative differential fibrations