The images of the higher generators via the evaluation map for the affine Yangian of type $A$
Mamoru Ueda
TL;DR
The paper addresses the problem of computing explicit images of higher generators under the evaluation map for the affine Yangian of type $A$, $Y_{\\hbar,\\varepsilon}(\\widehat{\\mathfrak{sl}}(n))$. It constructs the auxiliary algebra $y_{\\hbar}(\\mathfrak{gl}(n))$ and an embedding $\\iota$ of the subalgebra $Y_{\\hbar}(\\mathfrak{sl}(n))$ into it, together with an evaluation map $\\mathrm{ev}_{\\hbar}$ to $\\mathcal{U}(\\widehat{\\mathfrak{gl}}(n))$, and links this to the two-parameter evaluation map $\\mathrm{ev}_{\\hbar,\\varepsilon}$ via compatibility $\\mathrm{ev}_{\\hbar,\\varepsilon}\\circ\\kappa = \\mathrm{ev}_{\\hbar}\\circ\\iota$. The main contributions are explicit formulas for the images of higher generators $X^{\\pm}_{i,r}$ and $H_{i,r}$ (for $r\\ge2$) using symmetric polynomials $h_m$ and $f^m_n$, and demonstration of compatibility with GNW and related relations, clarifying the connection to rectangular $W$-algebras. Together, these results provide a concrete, computable bridge between affine Yangians and enveloping algebras of affine Lie algebras, enabling kernel descriptions and potential applications to $W$-algebra contexts.
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.