# Quantization of Poisson Hopf algebras

## TL;DR

This paper introduces a method to quantize Poisson Hopf algebras within symmetric monoidal categories, leveraging nerves of groups and Drinfeld associators to produce braided Hopf structures.

## Contribution

It presents a novel quantization technique compatible with tensor products that utilizes the nerve construction and Drinfeld associators for Poisson Hopf algebras.

## Key findings

- Quantization method compatible with tensor products.
- Nerves of Hopf algebras are braided, not symmetric.
- Poisson Hopf algebras become infinitesimally braided via this method.

## Abstract

We describe a method for quantization of Poisson Hopf algebras in $\mathbb Q$-linear symmetric monoidal categories. It is compatible with tensor products and can also be used to produce braided Hopf algebras. The main idea comes from the fact that nerves of groups are symmetric simplicial sets. Nerves of Hopf algebras then turn out to be braided rather than symmetric and nerves of Poisson Hopf algebras to be infinitesimally braided. The problem is thus solved via the standard machinery of Drinfeld associators.