# Nichols Algebras and Quantum Principal Bundles

## TL;DR

This paper develops a framework linking quantum principal bundles with Nichols algebras, applying it to quantum Grassmannians and conjecturing extensions to all quantum flag manifolds.

## Contribution

It introduces a novel framework connecting quantum principal bundles with Yetter-Drinfeld modules and Nichols algebras, specifically applied to quantum Grassmannians.

## Key findings

- Holomorphic and anti-holomorphic calculi expressed via Nichols algebras
- Framework applied to quantum Grassmannians
- Conjecture for all irreducible quantum flag manifolds

## Abstract

We introduce a general framework for associating to a homogeneous quantum principal bundle a Yetter-Drinfeld module structure on the cotangent space of the base calculus. The holomorphic and anti-holomorphic Heckenberger-Kolb calculi of the quantum Grassmannians are then presented in this framework. This allows us to express the calculi in terms of the corresponding Nichols algebras. The extension of this result to all irreducible quantum flag manifolds is then conjectured.