# Double Yangian and the universal R-matrix

## TL;DR

This paper provides a detailed algebraic construction of the double Yangian for rak{gl}_N, including the universal R-matrix, and proves foundational properties such as the PBW theorem and center description.

## Contribution

It offers a comprehensive, self-contained presentation of the double Yangian and its universal R-matrix, extending key structural results from the Yangian to the double Yangian.

## Key findings

- Construction of the universal R-matrix for the Yangian.
- Proof of the Poincare9-Birkhoff-Witt theorem for the double Yangian.
- Description of the center of the Yangian.

## Abstract

We describe the double Yangian of the general linear Lie algebra $\mathfrak{gl}_N$ by following a general scheme of Drinfeld. This description is based on the construction of the universal $R$-matrix for the Yangian. To make the exposition self contained, we include the proofs of all necessary facts about the Yangian itself. In particular, we describe the centre of the Yangian by using its Hopf algebra structure, and provide a proof of the analogue of the Poincar\'e-Birkhoff-Witt theorem for the Yangian based on its representation theory. This proof extends to the double Yangian, thus giving a description of its underlying vector space.