2018 Impact factor 4.843
Particles and Fields
Eur. Phys. J. C 7, 159-175
DOI 10.1007/s100529800968

Quantum orthogonal planes:
ISOq,r(N) and SOq,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 ISOq,r(N), and do contain dilatations. If we require bicovariance only under the quantum orthogonal group SOq,r(N), the calculus on the q-plane can be expressed in terms of its coordinates xa, differentials dxa 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 ISOq,r(n+1, m) and SOq,r(n+1,m) bicovariant calculi on the multiparametric quantum spaces. The particular case of the quantum Minkowski space ISOq,r(3,1)/SOq,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 xa and pa) are needed.


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