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Notes on a homomorphism from the affine Yangian associated with $\hat{\mathfrak{sl}}(n)$ to the affine Yangian associated with $\widehat{\mathfrak{sl}}(n+1)$

Mamoru Ueda

TL;DR

The paper investigates connections between the two-parameter affine Yangian $Y_{\hbar,\varepsilon}(\widehat{\mathfrak{sl}}(n))$ and non-rectangular $W$-algebras of type $A$ via degreewise completions and the maps $\Psi^{n,m}$, extending previous constructions to yield homomorphisms from affine Yangians to universal enveloping algebras of iterated $W$-algebras. It develops a network of explicit homomorphisms using compositions with $\Phi_u$, derives finite-analogs through reductions to finite $W$-algebras, and connects these to Li’s embeddings and Schur–Weyl duality, thereby enriching the algebraic realization of AGT-inspired correspondences. A key finding is the obstruction encountered when attempting to extend certain affine-Yangian homomorphisms directly to larger algebras in the non-rectangular setting, which prompts the proposal of a new, refined shifted affine Yangian. The results unify and extend several known embeddings between Yangians and $W$-algebras across affine and finite settings, offering a structural blueprint for future shifted-structure definitions and their representation-theoretic consequences.

Abstract

In the previous paper, we constructed a homomorphism from the affine Yangian associated with $\widehat{\mathfrak{sl}}(n)$ to the standard degreewise completion of the affine Yangian associated with $\widehat{\mathfrak{sl}}(n+1)$. In this article, by using this homomorphism, we construct homomorphisms from the affine Yangian to the universal enveloping algebra of a non-rectangular $W$-algebra of type $A$, which are different from the one in \cite{U7}. Based on this result, we discuss a new definition of the affine shifted Yangian, from which it is expected that there exists a homomorphism to the universal enveloping algebra of a non-rectangular $W$-algebra of type $A$. In the appendix, we also show that this homomorphism is related to the one from the quantum affine algebra associated with $\widehat{\mathfrak{sl}}(n)$ to the quantum affine algebra associated with $\widehat{\mathfrak{sl}}(n+1)$ given by Li.

Notes on a homomorphism from the affine Yangian associated with $\hat{\mathfrak{sl}}(n)$ to the affine Yangian associated with $\widehat{\mathfrak{sl}}(n+1)$

TL;DR

The paper investigates connections between the two-parameter affine Yangian and non-rectangular -algebras of type via degreewise completions and the maps , extending previous constructions to yield homomorphisms from affine Yangians to universal enveloping algebras of iterated -algebras. It develops a network of explicit homomorphisms using compositions with , derives finite-analogs through reductions to finite -algebras, and connects these to Li’s embeddings and Schur–Weyl duality, thereby enriching the algebraic realization of AGT-inspired correspondences. A key finding is the obstruction encountered when attempting to extend certain affine-Yangian homomorphisms directly to larger algebras in the non-rectangular setting, which prompts the proposal of a new, refined shifted affine Yangian. The results unify and extend several known embeddings between Yangians and -algebras across affine and finite settings, offering a structural blueprint for future shifted-structure definitions and their representation-theoretic consequences.

Abstract

In the previous paper, we constructed a homomorphism from the affine Yangian associated with to the standard degreewise completion of the affine Yangian associated with . In this article, by using this homomorphism, we construct homomorphisms from the affine Yangian to the universal enveloping algebra of a non-rectangular -algebra of type , which are different from the one in \cite{U7}. Based on this result, we discuss a new definition of the affine shifted Yangian, from which it is expected that there exists a homomorphism to the universal enveloping algebra of a non-rectangular -algebra of type . In the appendix, we also show that this homomorphism is related to the one from the quantum affine algebra associated with to the quantum affine algebra associated with given by Li.
Paper Structure (8 sections, 10 theorems, 100 equations)

This paper contains 8 sections, 10 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Affine Yangian
  3. $W$-algebras of type $A$
  4. Affine Yangians and non-rectangular $W$-algebras
  5. Finite Yangians and non-rectangular finite $W$-algebras of type $A$
  6. The embedding of the finite Yangian of type $A$
  7. Discussion of the new definition of the shifted affine Yangian
  8. Relationship between edge contractions for the quantum affine algebra and the finite Yangian

Key Result

Theorem 1.1

For $1\leq u\leq s$, we suppose that $q_u-q_{u+1}\geq3$ and $\dfrac{\varepsilon+q_u\hbar}{\hbar}=k+\sum_{v=1}^s\limits l_vq_v$. Then, we can obtain the homomorphism where $\mathcal{U}(\mathcal{W}^k(\mathfrak{gl}(\sum_{v=1}^s\limits l_vq_v),f_{q_1,q_2,\cdots,q_s}^{l_1,l_2\cdots,l_s}))$ means the universal enveloping algebra of a $W$-algebra $\mathcal{W}^k(\mathfrak{gl}(\sum_{v=1}^s\limits l_vq_v),

Theorems & Definitions (19)

  • Theorem 1.1
  • Definition 2.1: Definition 3.2 in Gu2 and Definition 2.1 in U8
  • Proposition 2.11
  • Theorem 2.22: Theorem 3.1 in U8
  • Definition 3.5
  • Theorem 3.6
  • proof
  • Remark 3.18
  • Theorem 3.19
  • Definition 4.2: Section 6 in MNT
  • ...and 9 more