The Geometry of the Quantum Euclidean Space

Abstract

A detailed study is made of the noncommutative geometry of $R_q^3$, the quantum space covariant under the quantum group $S{O}_q(3)$. For each of its two $S{O}_q(3)$-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.

29 pages, latex file

References in corpus (3)

• On Curvature in Noncommutative Geometry
• Quantum Space-time and Classical Gravity
• Differential Calculi and Linear Connections

Cited by in corpus (7)

• Unbraiding the braided tensor product
• On the Decoupling of the Homogeneous and Inhomogeneous Parts in Inhomogeneous Quantum Groups
• Twisted submanifolds of R^n
• On the hermiticity of q-differential operators and forms on the quantum Euclidean spaces R_q^N
• The fuzzy BTZ
• Twisted Quadrics and Algebraic Submanifolds in ${\mathbb{R}}^n$
• Frame formalism for the N-dimensional quantum Euclidean spaces