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A note on constructing quasi modules for quantum vertex algebras from twisted Yangians

Slaven Kožić, Marina Sertić

Abstract

In this note, we consider the twisted Yangians $\text{Y}(\mathfrak{g}_N)$ associated with the orthogonal and symplectic Lie algebras $\mathfrak{g}_N=\mathfrak{o}_N,\mathfrak{sp}_N$. First, we introduce a certain subalgebra $\text{A}_c(\mathfrak{g}_N)$ of the double Yangian for $\mathfrak{gl}_N$ at the level $c\in\mathbb{C}$, which contains the centrally extended $\text{Y}(\mathfrak{g}_N)$ at the level $c$ as well as its vacuum module $\mathcal{M}_c(\mathfrak{g}_N)$. Next, we employ its structure to construct examples of quasi modules for the quantum affine vertex algebra $\mathcal{V}_c(\mathfrak{gl}_N)$ associated with the Yang $R$-matrix. Finally, we use the description of the center of $\mathcal{V}_c(\mathfrak{gl}_N)$ to obtain explicit formulae for families of central elements for a certain completion of $\text{A}_c(\mathfrak{g}_N)$ and invariants of $\mathcal{M}_c(\mathfrak{g}_N)$.

A note on constructing quasi modules for quantum vertex algebras from twisted Yangians

Abstract

In this note, we consider the twisted Yangians associated with the orthogonal and symplectic Lie algebras . First, we introduce a certain subalgebra of the double Yangian for at the level , which contains the centrally extended at the level as well as its vacuum module . Next, we employ its structure to construct examples of quasi modules for the quantum affine vertex algebra associated with the Yang -matrix. Finally, we use the description of the center of to obtain explicit formulae for families of central elements for a certain completion of and invariants of .
Paper Structure (16 sections, 15 theorems, 98 equations)

This paper contains 16 sections, 15 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Double Yangian for glN
  4. Quantum affine vertex algebra
  5. Quasi modules for the quantum affine vertex algebra
  6. Twisted Yangians
  7. Vacuum module Mc(gN) as a quasi V2c(glN)-module
  8. Proof of Theorem \ref{['qmain']}
  9. Central elements in Ac(gN) and invariants in Mc(gN)
  10. Fusion procedure and the center of Vc(glN)
  11. Central elements of the completed algebra Ac(gN)
  12. Critical level
  13. Noncritical level
  14. Invariants of the vacuum module Mc(gN)
  15. Critical level
  16. ...and 1 more sections

Key Result

Theorem 2.1

For any $c\in \mathbb{C}\space$ there exists a unique structure of quantum vertex algebra on the $\mathbb{C}\space[[h]]$-module $\mathcal{V}_c(\mathfrak{gl}_N)$ such that the vacuum vector is the unit $\mathop{\mathrm{\boldsymbol{1}}}\in \mathcal{V}_c(\mathfrak{gl}_N)$ and the vertex operator map is

Theorems & Definitions (18)

  • Theorem 2.1
  • Remark 3.1
  • Proposition 3.2
  • Theorem 3.3
  • Remark 3.4
  • Corollary 3.5
  • Lemma 3.6
  • Lemma 3.7
  • Lemma 3.8
  • Lemma 4.1
  • ...and 8 more