On the uniqueness of Yangians

TL;DR

This paper proves the uniqueness of the Yangian as a homogeneous quantization of the polynomial current algebra of a simple Lie algebra, filling a gap in the foundational understanding of Yangians.

Contribution

It provides a complete proof of the uniqueness of the Yangian quantization, combining cohomological and computational methods, and offers a simplified presentation using Drinfeld's generators.

Findings

Proof that Yangians are uniquely determined as homogeneous quantizations.

A presentation of Yangians with a reduced set of defining relations.

Confirmation of Drinfeld's original characterization of Yangians.

Abstract

Let $\mathfrak{g}$ be a simple Lie algebra over the complex numbers, and let $\mathfrak{g}[u]$ denote its polynomial current algebra. In the mid-1980s, Drinfeld introduced the Yangian of $\mathfrak{g}$ as the unique solution to a quantization problem for a natural Lie bialgebra structure on $\mathfrak{g}[u]$. More precisely, Theorem 2 of [Dokl. Akad. Nauk SSSR 283 (1985), no. 5, 1060-1064] asserts that $\mathfrak{g}[u]$ admits a unique homogeneous quantization, the Yangian of $\mathfrak{g}$, which is described explicitly via generators and relations, starting from a copy of $\mathfrak{g}$ and its adjoint representation. Although the representation theory of Yangians has since undergone substantial development, a complete proof of Drinfeld's theorem has not appeared. In this article, we present a proof of the assertion that $\mathfrak{g}[u]$ admits at most one homogeneous quantization. Our argument combines cohomological and computational methods, and outputs a presentation of any such quantization using Drinfeld's generators and a reduced set of defining relations.