Geometrical techniques for the N-dimensional Quantum Euclidean Spaces

2000/02/25 by B. L. Cerchiai, Bianca L. Cerchiai, G. Fiore +5
Mathematics · Physics and Astronomy · #17B37 #81R50 #Advanced Operator Algebra Research #Black Holes and Theoretical Physics #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Noncommutative and Quantum Gravity Theories #Quantum Algebra (math.QA) #math-ph #math.MP #math.QA #msc:17B37 #msc:81R50

paper · pdf · doi:10.48550/arxiv.math/0002215

latex, 11 pages, talk given at the VI Wigner Symposium

arxiv created 2000/02/25 · openalex publication_date 2000/02/25 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We briefly report our application of a version of noncommutative geometry to the quantum Euclidean space R^N_q, for any N ≥ 3; this space is covariant under the action of the quantum group SO_q(N), and two covariant differential calculi are known on it. More precisely, we describe how to construct in a Cartan `moving-frame formalism' the metric, two covariant derivatives, the Dirac operator, the frame, the inner derivations dual to the frame elements, for both of these calculi. The components of the frame elements in the basis of differentials provide a `local realization' of the Faddeev-Reshetikhin-Takhtadjan generators of U_q^±(so(N)).

Related

• Cartan Calculus on the Quantum Space \cal R_q³
• Metrics on the Real Quantum Plane
• Noncommutative quantum analogs of constant curvature spaces
• Quantum Euclidean Spaces with Noncommutative Derivatives
• Quantum Riemannian geometry of phase space and nonassociativity