The Geometry of the Quantum Euclidean Space

Details The Geometry of the Quantum Euclidean Space The Geometry of the Quantum Euclidean Space

1999-04-06

arxiv.org/abs/math/9904027v1

Mathematics

Quantum Algebra

29 pages, latex file

Scientific paper

10.1016/S0393-0440(99)00054-6

A detailed study is made of the noncommutative geometry of $R^3_q$, the quantum space covariant under the quantum group $SO_q(3)$. For each of its two $SO_q(3)$-covariant differential calculi we find its metric, the corresponding frame and two torsion-free covariant derivatives that are metric compatible up to a conformal factor and which yield both a vanishing linear curvature. A discussion is given of various ways of imposing reality conditions. The delicate issue of the commutative limit is discussed at the formal algebraic level. Two rather different ways of taking the limit are suggested, yielding respectively $S^2\times R$ and $R^3$ as the limit Riemannian manifold.

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