Abstract:
We show that the crossed modules and bicovariant differential calculi on two Hopf algebras related by a cocycle twist are in 1-1 correspondence. In particular, for quantum groups which are cocycle deformation-quantisations of classical groups the calculi are obtained as deformation-quantisations of the classical ones. As an application, we classify all bicovariant differential calculi on the Planck scale Hopf algebra . This is a quantum group which has an limit as the functions on a classical but non-Abelian group and a limit as flat space quantum mechanics. We further study the noncommutative differential geometry and Fourier theory for this Hopf algebra as a toy model for Planck scale physics. The Fourier theory implements a T-duality-like self-duality. The noncommutative geometry turns out to be singular when and is therefore not visible in flat space quantum mechanics alone.
Similar content being viewed by others
Author information
Authors and Affiliations
Additional information
Received: 28 October 1998 / Accepted: 7 March 1999
Rights and permissions
About this article
Cite this article
Majid, S., Oeckl, R. Twisting of Quantum Differentials and¶the Planck Scale Hopf Algebra. Comm Math Phys 205, 617–655 (1999). https://doi.org/10.1007/s002200050692
Issue Date:
DOI: https://doi.org/10.1007/s002200050692