An Example of Homomorphisms from the Guay's affine Yangians to non-rectangular $W$-algebras

TL;DR

This paper constructs a non-trivial algebra homomorphism linking Guay's affine Yangians to certain $W$-algebras, advancing understanding of their algebraic relationships in representation theory.

Contribution

It introduces a novel homomorphism from Guay's affine Yangian to a specific $W$-algebra associated with a nilpotent element in $rak{gl}(m+n)$.

Findings

Established a new algebra homomorphism between affine Yangians and $W$-algebras.

Connected the structure of Guay's affine Yangians with $W$-algebras for specific nilpotent elements.

Provided algebraic tools for further exploration of representation theory relationships.

Abstract

We construct a non-trivial homomorphism from the Guay's affine Yangian associated with $\widehat{\mathfrak{sl}}(n)$ to the universal enveloping algebra of the $W$-algebra associated with a Lie algebra $\mathfrak{gl}(m+n)$ and its nilpotent element whose Jordan block of type $(2^{n},1^{m-n})$ for $m>n$.