# Poisson Principal Bundles

## TL;DR

This paper develops a semiclassical framework for quantum group principal bundles within Poisson geometry, describing the structures of total space, fiber, and base with compatible Poisson connections, and illustrates it with the q-Hopf fibration.

## Contribution

It introduces a Poisson geometric analogue of quantum principal bundles, including connections and examples like the q-Hopf fibration and spin connection.

## Key findings

- Established Poisson-compatible contravariant connections for total space, fiber, and base.
- Constructed the Poisson level of the q-Hopf fibration on the standard q-sphere.
- Developed the Poisson level of the spin connection on principal bundles.

## 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.