Coproduct for affine Yangians and parabolic induction for rectangular $W$-algebras

TL;DR

This paper constructs algebra homomorphisms from affine Yangians to rectangular W-algebras, enabling a new perspective on parabolic induction as tensor products, with applications to superalgebras.

Contribution

It introduces a coproduct-based construction of homomorphisms from affine Yangians to rectangular W-algebras, linking parabolic induction to tensor products.

Findings

Homomorphisms from affine Yangians to rectangular W-algebras constructed

Parabolic induction corresponds to tensor product representations

Method extends to superalgebra setting

Abstract

We construct algebra homomorphisms from affine Yangians to the current algebras of rectangular $W$-algebras both in type A. The construction is given via the coproduct and the evaluation map for the affine Yangians. As a consequence, we show that parabolic inductions for representations of the rectangular $W$-algebras can be regarded as tensor product representations of the affine Yangians under the homomorphisms. The same method is applicable also to the super setting.