# Characteristic (Fedosov-)class of a twist constructed by Drinfel'd

## TL;DR

This paper connects Drinfel'd's twist construction with Fedosov quantization by computing the characteristic class, showing it is trivial and related to the symplectic form divided by Planck's constant.

## Contribution

It establishes a link between Drinfel'd's classical r-matrix-based twist and Fedosov's quantization, explicitly computing the characteristic class of the twist.

## Key findings

- The characteristic Fedosov class of the Drinfel'd twist is trivial.
- The class is given by the symplectic form divided by .
- This unifies two approaches to deformation quantization.

## Abstract

In a seminal paper Drinfel'd explained how to associate to every classical r-matrix for a Lie algebra $\lie g$ a twisting element based on $\mathcal{U}(\lie g)[[\hbar]]$, or equivalently a left invariant star product of the corresponding symplectic structure $\omega$ on the 1-connected Lie group G of g. In a recent paper, the authors solve the same problem by means of Fedosov quantization. In this short note we provide a connection between the two constructions by computing the characteristic (Fedosov) class of the twist constructed by Drinfel'd and proving that it is the trivial class given by $ \frac{[\omega]}{\hbar}$.