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Combinatorial twists in gl_n Yangians

Anastasia Doikou

TL;DR

The paper develops a framework to produce combinatorial set-theoretic solutions of the Yang-Baxter equation via universal Drinfeld twists in the $\mathfrak{gl}_n$ Yangian setting. It constructs a special set-theoretic YB algebra as a Hopf algebra, derives an admissible set-theoretic twist ${\cal F}$, and obtains the twisted universal ${\cal R}$-matrix ${\cal R}^F$, illustrating how brace-like structures arise from twisting. It then extends these ideas to the gl$_n$ Yangian ${\cal Y}(\mathfrak{gl}_n)$ and to the augmented Yangian ${\cal Y}_n^+$, establishing their Hopf algebra structures and producing twisted versions ${\cal Y}_n^{+F}$ with corresponding twisted $L$-operators and $R$-matrices. The results provide a concrete algebraic bridge between braces (and skew braces) and quantum groups, yielding explicit set-theoretic ${\cal R}$-matrices represented by $n^2\times n^2$ matrices and parametrized by spectral parameters, with potential applications to parametrized YBE solutions and braided Hopf-algebraic frameworks.

Abstract

We introduce the special set-theoretic Yang-Baxter algebra and show that it is a Hopf algebra subject to certain conditions. The associated universal R-matrix is also obtained via an admissible Drinfel'd twist. The structure of braces emerges naturally in this context by requiring the special set-theoretic Yang-Baxter algebra to be a Hopf algebra and a quasi-triangular bialgebra after twisting. The fundamental representation of the universal R-matrix yields the familiar set-theoretic (combinatorial) solutions of the Yang-Baxter equation. We then apply the same Drinfel'd twist to the gl_n Yangian after introducing the augmented Yangian. We show that the augmented Yangian is also a Hopf algebra and we also obtain its twisted version.

Combinatorial twists in gl_n Yangians

TL;DR

The paper develops a framework to produce combinatorial set-theoretic solutions of the Yang-Baxter equation via universal Drinfeld twists in the Yangian setting. It constructs a special set-theoretic YB algebra as a Hopf algebra, derives an admissible set-theoretic twist , and obtains the twisted universal -matrix , illustrating how brace-like structures arise from twisting. It then extends these ideas to the gl Yangian and to the augmented Yangian , establishing their Hopf algebra structures and producing twisted versions with corresponding twisted -operators and -matrices. The results provide a concrete algebraic bridge between braces (and skew braces) and quantum groups, yielding explicit set-theoretic -matrices represented by matrices and parametrized by spectral parameters, with potential applications to parametrized YBE solutions and braided Hopf-algebraic frameworks.

Abstract

We introduce the special set-theoretic Yang-Baxter algebra and show that it is a Hopf algebra subject to certain conditions. The associated universal R-matrix is also obtained via an admissible Drinfel'd twist. The structure of braces emerges naturally in this context by requiring the special set-theoretic Yang-Baxter algebra to be a Hopf algebra and a quasi-triangular bialgebra after twisting. The fundamental representation of the universal R-matrix yields the familiar set-theoretic (combinatorial) solutions of the Yang-Baxter equation. We then apply the same Drinfel'd twist to the gl_n Yangian after introducing the augmented Yangian. We show that the augmented Yangian is also a Hopf algebra and we also obtain its twisted version.
Paper Structure (9 sections, 11 theorems, 53 equations)

This paper contains 9 sections, 11 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Set-theoretic Hopf algebras
  3. Special set-theoretic YB algebra as a Hopf algebra
  4. Set-theoretic Drinfel'd twist
  5. The twisted Hopf algebra
  6. Twisting the gl_n Yangian
  7. Preliminaries: a review on the gl_n Yangian
  8. The Yangian Y(gl_n)
  9. Twisting the Yangian

Key Result

proposition 1

Let ${\cal A}$ be the basic set-theoretic YB algebra then If, in addition, for all $a,b \in X,$$h_a = h_b \Rightarrow a =b,$ then for all $a,b,c \in X,$ and $\sigma_a: X \to X$ is an injection.

Theorems & Definitions (39)

  • definition 1
  • definition 2
  • remark 1
  • definition 3
  • proposition 1
  • proof
  • definition 4
  • remark 2
  • remark 3: Fundamental representation of the special set-theoretic YB algebra:
  • theorem 1: Hopf algebra
  • ...and 29 more