NONCOMMUTATIVE RIEMANNIAN AND SPIN GEOMETRY OF THE STANDARD q-SPHERE
نویسنده
چکیده
We study the quantum sphere Cq [S] as a quantum Riemannian manifold in the quantum frame bundle approach. We exhibit its 2-dimensional cotangent bundle as a direct sum Ω ⊕ Ω in a double complex. We find the natural metric, volume form, Hodge * operator, Laplace and Maxwell operators. We show that the q-monopole as spin connection induces a natural Levi-Civita type connection and find its Ricci curvature and q-Dirac operator ∇/. We find the possibility of an antisymmetric volume form quantum correction to the Ricci curvature and Lichnerowicz-type formulae for ∇/. We also remark on the geometric q-Borel-Weil-Bott construction.
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