The $R$-matrix presentation for the Yangian of a simple Lie algebra

TL;DR

This paper constructs an extended Yangian algebra from a finite-dimensional representation of the Yangian of a simple Lie algebra, using an $R$-matrix presentation, and proves structural properties including a central series and a surjective morphism to the original Yangian.

Contribution

It introduces the extended Yangian $X_ ext{I}(rak{g})$ with relations encoded by an $R$-matrix, and establishes its connection to the standard Yangian, generalizing classical results.

Findings

Constructed the extended Yangian $X_ ext{I}(rak{g})$ from a finite-dimensional representation.

Proved the existence of a surjective Hopf algebra morphism to the Yangian.

Demonstrated that the central matrix $ ext{Z}(u)$ becomes grouplike when the representation is irreducible.

Abstract

Starting from a finite-dimensional representation of the Yangian $Y(\mathfrak{g})$ for a simple Lie algebra $\mathfrak{g}$ in Drinfeld's original presentation, we construct a Hopf algebra $X_\mathcal{I}(\mathfrak{g})$, called the extended Yangian, whose defining relations are encoded in a ternary matrix relation built from a specific $R$-matrix $R(u)$. We prove that there is a surjective Hopf algebra morphism $X_\mathcal{I}(\mathfrak{g})\twoheadrightarrow Y(\mathfrak{g})$ whose kernel is generated as an ideal by the coefficients of a central matrix $\mathcal{Z}(u)$. When the underlying representation is irreducible, we show that this matrix becomes a grouplike central series, thereby making available a proof of a well-known theorem stated by Drinfeld in the 1980's. We then study in detail the algebraic structure of the extended Yangian, and prove several generalizations of results which are known to hold for Yangians associated to classical Lie algebras in their $R$-matrix presentations.