On Guay's evaluation map for affine Yangians

TL;DR

This paper provides a detailed proof of the evaluation map for affine Yangians of type A, clarifying parameter assumptions and determining highest weights of evaluation modules, thus filling gaps in the existing literature.

Contribution

It offers the first rigorous proof of the evaluation map’s well-definedness and specifies the necessary parameter assumptions, advancing understanding of affine Yangian representations.

Findings

Proof of the evaluation map’s existence and well-definedness

Explicit determination of highest weights of evaluation modules

Clarification of parameter assumptions needed for the map

Abstract

We give a detailed proof of the existence of evaluation map for affine Yangians of type A to clarify that it needs an assumption on parameters. This map was first found by Guay but a proof of its well-definedness and the assumption have not been written down in the literature. We also determine the highest weights of evaluation modules defined as the pull-back of integrable highest weight modules of the affine Lie algebra $\hat{\mathfrak{gl}}_N$ by the evaluation map.