Commuting Subalgebras of Affine Super Yangians Arising from Edge Contractions

TL;DR

This paper explores the structure of affine super Yangians through edge contractions, establishing their compatibility with algebraic operations and constructing homomorphisms to related algebraic structures.

Contribution

It introduces two commuting edge contractions for affine super Yangians and demonstrates their compatibility with coproducts and homomorphisms to $W$-superalgebras.

Findings

Edge contractions commute with each other.

Constructed homomorphisms to centralizer algebras.

Established compatibility with coproducts and parabolic induction.

Abstract

In the previous paper, we constructed two kinds of edge contractions for the affine super Yangian and a homomorphism from the affine super Yangian to the universal enveloping algebra of a $W$-superalgebra of type $A$. In this article, we show that these two edge contractions commute with each other. As an application, we give a homomorphism from the affine super Yangian to some centralizer algebras of the universal enveloping algebra of $W$-superalgebras of type $A$. Using the edge contraction, we also show the compatibility of the coproduct for the affine super Yangian with the parabolic induction for a $W$-superalgebra of type $A$ in some special cases.