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Yangian deformations of $\mathcal{S}$-commutative quantum vertex algebras and Bethe subalgebras

Lucia Bagnoli, Slaven Kožić

Abstract

We construct a new class of quantum vertex algebras associated with the normalized Yang $R$-matrix. They are obtained as Yangian deformations of certain $\mathcal{S}$-commutative quantum vertex algebras and their $\mathcal{S}$-locality takes the form of a single $RTT$-relation. We establish some preliminary results on their representation theory and then further investigate their braiding map. In particular, we show that its fixed points are closely related with Bethe subalgebras in the Yangian quantization of the Poisson algebra $\mathcal{O}(\mathfrak{gl}_N((z^{-1})))$, which were recently introduced by Krylov and Rybnikov. Finally, we extend this construction of commutative families to the case of trigonometric $R$-matrix of type $A$.

Yangian deformations of $\mathcal{S}$-commutative quantum vertex algebras and Bethe subalgebras

Abstract

We construct a new class of quantum vertex algebras associated with the normalized Yang -matrix. They are obtained as Yangian deformations of certain -commutative quantum vertex algebras and their -locality takes the form of a single -relation. We establish some preliminary results on their representation theory and then further investigate their braiding map. In particular, we show that its fixed points are closely related with Bethe subalgebras in the Yangian quantization of the Poisson algebra , which were recently introduced by Krylov and Rybnikov. Finally, we extend this construction of commutative families to the case of trigonometric -matrix of type .
Paper Structure (8 sections, 14 theorems, 160 equations)

This paper contains 8 sections, 14 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. On certain algebras associated with the Yang $R$-matrix
  4. Quantum vertex algebra V(R)
  5. On V(R)-modules
  6. Fixed points of the braiding
  7. Commutative families in the Yangian quantization of $\mathcal{O}(\mathfrak{gl}_N((z^{-1})))$
  8. Trigonometric case

Key Result

Proposition 3.1

The assignments define a structure of $A(R^{\pm})$-module on $\mathcal{P}^\pm$. Moreover, the images of the monomials are such that $i_1\leqslant\ldots\leqslant i_m$ (resp. $i_1<\ldots< i_m$) and $j_1\leqslant\ldots\leqslant j_m$ (resp. $j_1<\ldots< j_m$), form a (topological) basis of the image of $A(R^{+})$ (resp. $A(R^{-})$) under this representation.

Theorems & Definitions (18)

  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Definition 4.1
  • Theorem 4.2
  • Definition 5.1
  • Corollary 5.2
  • Proposition 5.3
  • Proposition 6.1
  • Proposition 6.2
  • ...and 8 more