Liam Williams | Searcharxiv

arXiv2021

Poisson Principal Bundles

We semiclassicalise the theory of quantum group principal bundles to the level of Poisson geometry. The total space $X$ is a Poisson manifold with Poisson-compatible contravariant connection, the fibre is a Poisson-Lie group in the sense of Drinfeld with bicovariant Poisson-compatible contravariant connection, and the base has an inherited Poisson structure and Poisson-compatible contravariant connection. The latter are known to be the semiclassical data for a quantum differential calculus. The theory is illustrated by the Poisson level of the $q$-Hopf fibration on the standard $q$-sphere. We also construct the Poisson level of the spin connection on a principal bundle.

math.QA↗

arXiv2017

Quantum Koszul formula on quantum spacetime

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 $Ω^1\otimes_AΩ^1\to A$ where $A$ is a possibly noncommutative or `quantum' spacetime coordinate algebra and $(Ω^1,d)$ is a specified bimodule of 1-forms or `differential calculus' over it. In this paper we explore the proposal of a `quantum Koszul formula' with initial data a degree -2 bilinear map $\perp$ on the full exterior algebra $Ω$ obeying the 4-term relations \[ (-1)^{|η|} (ωη)\perpζ+(ω\perpη)ζ=ω\perp(ηζ)+(-1)^{|ω|+|η|}ω(η\perpζ),\quad\forallω,η,ζ\inΩ\] and a compatible degree -1 `codifferential' map $δ$. These provide a quantum metric and interior product and a canonical bimodule connection $\nabla$ on all degrees. The theory is also more general than classically in that we do not assume symmetry of the metric nor that $δ$ is obtained from the metric. We solve and interpret the $(δ,\perp)$ data on the bicrossproduct model quantum spacetime $[r,t]=λr$ for its two standard choices of $Ω$. For the $α$-family calculus the construction includes the quantum Levi-Civita connection for a general quantum symmetric metric, while for the more standard $β=1$ calculus we find the quantum Levi-Civita connection for a quantum `metric' that in the classical limit is antisymmetric.

math.QA↗