Quantum orthogonal planes: $ISO_{q,r}(N)$ and $SO_{q,r}(N)$ – bicovariant calculi and differential geometry on quantum Minkowski space

P. Aschieri; L. Castellani; A.M. Scarfone

doi:doi:10.1007/s100529800968

2021 Impact factor 4.991

Particles and Fields

Eur. Phys. J. C 7, 159-175
DOI 10.1007/s100529800968

Quantum orthogonal planes:
ISO_q,r(N) and SO_q,r(N) - bicovariant calculi
and differential geometry on quantum Minkowski space

P. Aschieri¹ - L. Castellani² - A.M. Scarfone³

¹ Theoretical Physics Group, Physics Division, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley,
California 94720, USA (e-mail: aschieri@lbl.gov)
² Dipartimento di Scienze e Tecnologie Avanzate^*, Università di Torino and Dipartimento di Fisica Teorica and Istituto Nazionale di Fisica Nucleare, Via P. Giuria 1, I-10125 Torino, Italy (e-mail: castellani@to.infn.it)
³ Dipartimento di Fisica, Politecnico di Torino, Corso Duca degli Abruzzi 24, I-10129 Torino, Italy (e-mail: scarfone@polito.it)

Received: 4 November 1997 / Revised version: 22 April 1998 / Published online: 3 December 1998

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.