paper

Quantum Orthogonal Planes: ISO_{q,r}(N) and SO_{q,r}(N) -- Bicovariant Calculi and Differential Geometry on Quantum Minkowski Space

arXiv:q-alg/9709032 · doi:10.1007/s100529800968

Abstract

We construct differential calculi on multiparametric quantum orthogonal planes in any dimension N. These calculi are bicovariant under the action of the full inhomogeneous (multiparametric) quantum group ISO_{q,r}(N), and do contain dilatations. If we require bicovariance only under the quantum orthogonal group SO_{q,r}(N), the calculus on the q-plane can be expressed in terms of its coordinates x^a, differentials dx^a and partial derivatives \partial_a without the need of dilatations, thus generalizing known results to the multiparametric case. Using real forms that lead to the signature (n+1,m) with m = n-1, n, n+1 , we find ISO_{q,r}(n+1, m) and SO_{q,r}(n+1,m) bicovariant calculi on the multiparametric quantum spaces. The particular case of the quantum Minkowski space ISO_{q,r}(3,1)/SO_{q,r}(3,1) is treated in detail. The conjugated partial derivatives \partial_a* can be expressed as linear combinations of the \partial_a. This allows a deformation of the phase-space where no additional operators (besides x^a and p_a) are needed.

LaTeX, 36 pages. Considered more real forms, added some explicit formulas, used simpler definition of hermitean momenta. To be published in European Phys. Jou. C

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