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Commuting Subalgebras of Affine Super Yangians Arising from Edge Contractions

Mamoru Ueda

TL;DR

This work establishes that two edge contractions on the affine super Yangian, denoted as $Psi1$ and $Psi2$, commute, enabling a tensor product construction into a larger completed affine super Yangian. It then connects the affine super Yangian to centralizer algebras of universal enveloping algebras of $W$-superalgebras of type A via evaluation maps, and demonstrates the compatibility of the Yangian coproduct with parabolic induction in special cases. A bridge is developed between affine super Yangians and $W$-superalgebras through a family of commuting homomorphisms $Phi_s$, whose images land in centralizers of $W$-algebras and reflect the structure of parabolic inductions. The paper also introduces an extended affine super Yangian and a corresponding coproduct-like map, expanding the Hopf-like framework. In the rectangular and special settings, Miura maps are leveraged to realize these algebras inside tensor products of vertex algebras, enabling a cohesive picture that ties together edge contractions, centralizers, and parabolic inductions, with potential connections to generalized trialities of $Y$-algebras.

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.

Commuting Subalgebras of Affine Super Yangians Arising from Edge Contractions

TL;DR

This work establishes that two edge contractions on the affine super Yangian, denoted as and , commute, enabling a tensor product construction into a larger completed affine super Yangian. It then connects the affine super Yangian to centralizer algebras of universal enveloping algebras of -superalgebras of type A via evaluation maps, and demonstrates the compatibility of the Yangian coproduct with parabolic induction in special cases. A bridge is developed between affine super Yangians and -superalgebras through a family of commuting homomorphisms , whose images land in centralizers of -algebras and reflect the structure of parabolic inductions. The paper also introduces an extended affine super Yangian and a corresponding coproduct-like map, expanding the Hopf-like framework. In the rectangular and special settings, Miura maps are leveraged to realize these algebras inside tensor products of vertex algebras, enabling a cohesive picture that ties together edge contractions, centralizers, and parabolic inductions, with potential connections to generalized trialities of -algebras.

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 -superalgebra of type . 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 -superalgebras of type . Using the edge contraction, we also show the compatibility of the coproduct for the affine super Yangian with the parabolic induction for a -superalgebra of type in some special cases.
Paper Structure (14 sections, 22 theorems, 131 equations)

This paper contains 14 sections, 22 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Affine super Yangian
  3. Edge contractions for the affine super Yangian
  4. Affine super Yangians and centralizer algebras of $\boldsymbol{U\bigl(\widehat{\mathfrak{gl}}(n)\bigr)}$
  5. $\boldsymbol{W}$-superalgebras of type $\boldsymbol{A}$
  6. Affine super Yangians and $\boldsymbol{W}$-superalgebras of type $\boldsymbol{A}$
  7. Extended affine super Yangian
  8. Compatibility of $\boldsymbol{\Phi_l}$ with the parabolic induction for a $\boldsymbol{W}$-superalgebra in the special setting
  9. Some formulas for Theorem \ref{['Commutativity']}
  10. Proof of Theorem \ref{['Commutativity']}
  11. The proof of (\ref{['gather1']})
  12. The proof of (\ref{['gather2']})
  13. The proof of (\ref{['gather3']})
  14. Proof of Theorem \ref{['Tinf']}

Key Result

Theorem 1.1

The images of $\Psi_1$ and $\Psi_2$ commute with each other.

Theorems & Definitions (30)

  • Theorem 1.1
  • Definition 2.1: U2
  • Theorem 2.2: U4
  • Theorem 2.3: U2
  • Theorem 3.1: U13
  • Theorem 3.2
  • Theorem 4.1: U2 and U3
  • Theorem 4.2
  • Remark 5.1
  • Definition 5.2
  • ...and 20 more