Quantum Koszul formula on quantum spacetime
Journal of Geometry and Physics
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Noncommutative or quantum Riemannian geometry has been proposed as an effective theory for aspects of quantum gravity. Here the metric is an invertible bimodule map where is a possibly noncommutative or ‘quantum’ spacetime coordinate algebra and is a specified bimodule of 1-forms or ‘differential calculus’ over it. In this paper we explore the proposal of a ‘quantum Koszul formula’ in Majid  with initial data a degree bilinear map on the full exterior algebra obeying the 4-term relations and a compatible degree ‘codifferential’ map . These provide a quantum metric, interior product and a canonical bimodule connection on all degrees. The theory is also more general than classically in that we do not assume symmetry of the metric nor that is obtained from the metric. We solve and interpret the data on the bicrossproduct model quantum spacetime for its two standard choices of . For the -family calculus the construction includes the quantum Levi-Civita connection for a general quantum symmetric metric, while for the more standard calculus we find the quantum Levi-Civita connection for a quantum ‘metric’ that in the classical limit is antisymmetric. This suggests to consider quantum Riemannian and symplectic geometry on a more equal footing than is currently the case.
AuthorsMAJID, SH; WILLIAMS, L
- Mathematics