The coproduct for the affine Yangian and the parabolic induction for non-rectangular $W$-algebras

TL;DR

This paper constructs a homomorphism linking the affine Yangian and non-rectangular W-algebras, demonstrating compatibility of the coproduct with parabolic induction and exploring the image in affine cosets.

Contribution

It introduces a new homomorphism connecting affine Yangians and non-rectangular W-algebras, extending previous work and analyzing its properties.

Findings

Coproduct for affine Yangian is compatible with parabolic induction.

Homomorphism maps affine Yangian to non-rectangular W-algebras.

Image of the homomorphism lies in the affine coset in specific cases.

Abstract

By using the coproduct and evaluation map for the affine Yangian and the Miura map for non-rectangular $W$-algebras, we construct a homomorphism from the affine Yangian associated with $\widehat{\mathfrak{sl}}(n)$ to the universal enveloping algebra of a non-rectangular $W$-algebra of type $A$, which is an affine analogue of the one given in De Sole-Kac-Valeri. As a consequence, we find that the coproduct for the affine Yangian is compatible with some of the parabolic induction for non-rectangular $W$-algebras via this homomorphism. We also show that the image of this homomorphism is contained in the affine coset of the $W$-algebra in the special case that the $W$-algebra is associated with a nilpotent element of type $(1^{m-n},2^n)$.