The images of the higher generators via the evaluation map for the affine Yangian of type $A$

TL;DR

This paper explicitly computes the images of higher generators under the evaluation map for the affine Yangian of type A, clarifying their structure beyond finitely many known generators.

Contribution

It provides explicit formulas for the images of higher generators in the current presentation of the affine Yangian of type A, extending previous understanding.

Findings

Explicit images of higher generators obtained

Clarifies the structure of the evaluation map

Enhances understanding of affine Yangian representations

Abstract

The affine Yangian associated with $\widehat{\mathfrak{sl}}(n)$ has several presentations: the current presentation, the minimalistic presentation and so on. The evaluation map for the affine Yangian was given by using the minimalistic presentation. One of the issues about the evaluation map is that the images of the evaluation maps are unkown except on finitely many generators. In this article, we write down the images of the higher generators of the current presentation via the evaluation map for the affine Yangian of type $A$ explicitly.