The Noncommutative Geometry of the Quantum Projective Plane

Abstract

We study the spectral geometry of the quantum projective plane CP^2_q, a deformation of the complex projective plane CP^2, the simplest example of a spin^c manifold which is not spin. In particular, we construct a Dirac operator D which gives a 0^+ summable spectral triple, equivariant under U_q(su(3)). The square of D is a central element for which left and right actions on spinors coincide, a fact that is exploited to compute explicitly its spectrum.

v2: 26 pages. Paper completely reorganized; no major change, several minor ones

References in corpus (5)

• The Dirac operator on SU_q(2)
• A Local Index Formula for the Quantum Sphere
• Projective quantum spaces
• The Isospectral Dirac Operator on the 4-dimensional Orthogonal Quantum Sphere
• Non integrable representations of the restricted quantum analogue of sl(3) at roots of 1

Cited by in corpus (11)

• Dirac Operators on Quantum Projective Spaces
• Anti-selfdual Connections on the Quantum Projective Plane: Monopoles
• Noncommutative complex geometry of the quantum projective space
• The homogeneous coordinate ring of the quantum projective plane
• Liberations and twists of real and complex spheres
• Quantum Bundle Description of the Quantum Projective Spaces
• Quantum weighted projective and lens spaces
• Anti-selfdual Connections On The Quantum Projective Plane: Instantons
• Quantum Dimension and Quantum Projective Spaces
• Dolbeault-Dirac operators, quantum Clifford algebras and the Parthasarathy formula
• Quantum Riemannian geometry of quantum projective spaces