Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 006, 23 pages      arXiv:1903.12006      https://doi.org/10.3842/SIGMA.2021.006
Contribution to the Special Issue on Noncommutative Manifolds and their Symmetries in honour of Giovanni Landi

Poisson Principal Bundles

Shahn Majid and Liam Williams
School of Mathematical Sciences, Queen Mary University of London, Mile End Rd, London E1 4NS, UK

Received June 11, 2020, in final form January 05, 2021; Published online January 13, 2021

Abstract
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.

Key words: noncommutative geometry; quantum group; gauge theory; symplectic geometry; poisson geometry; Lie bialgebra; homogenous space; $q$-monopole.

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