Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Algebraic structures in set-theoretic Yang-Baxter & reflection equations

Anastasia Doikou

TL;DR

The paper develops an algebraic framework for invertible, non-degenerate set-theoretic solutions of the Yang-Baxter and reflection equations using braces and near-braces, deriving a quantum algebra via RTT and connecting brace-based R-matrices to twisted Yangians through a Drinfeld twist. It Baxterizes involutive solutions to obtain $R()$-matrices and analyzes the resulting L-operator algebras, including an explicit twist that relates brace-derived structures to the Yangian. It extends the construction to the set-theoretic reflection equation, formulating the reflection algebra from Baxterized $R$-matrices and identifying finite subalgebras for special reflection maps. These results provide a systematic, algebraic pathway to integrable models with boundaries and set the stage for diagonalization and universal $R$-matrices in this combinatorial setting.

Abstract

We present resent results regarding invertible, non-degenerate solutions of the set-theoretic Yang-Baxter and reflection equations. We recall the notion of braces and we present and prove various fundamental properties required for the solutions of the set theoretic Yang-Baxter equation. We then restrict our attention on involutive solutions and consider lambda parametric set-theoretic solutions of the Yang-Baxter equation and we extract the associated quantum algebra. We also discuss the notion of the Drinfeld twist for involutive solutions and their relation to the Yangian. We next focus on reflections and we derive the associated defining algebra relations for R-matrices being Baxterized solutions of the symmetric group. We show that there exists a ``reflection'' finite sub-algebra for some special choice of reflection maps.

Algebraic structures in set-theoretic Yang-Baxter & reflection equations

TL;DR

The paper develops an algebraic framework for invertible, non-degenerate set-theoretic solutions of the Yang-Baxter and reflection equations using braces and near-braces, deriving a quantum algebra via RTT and connecting brace-based R-matrices to twisted Yangians through a Drinfeld twist. It Baxterizes involutive solutions to obtain -matrices and analyzes the resulting L-operator algebras, including an explicit twist that relates brace-derived structures to the Yangian. It extends the construction to the set-theoretic reflection equation, formulating the reflection algebra from Baxterized -matrices and identifying finite subalgebras for special reflection maps. These results provide a systematic, algebraic pathway to integrable models with boundaries and set the stage for diagonalization and universal -matrices in this combinatorial setting.

Abstract

We present resent results regarding invertible, non-degenerate solutions of the set-theoretic Yang-Baxter and reflection equations. We recall the notion of braces and we present and prove various fundamental properties required for the solutions of the set theoretic Yang-Baxter equation. We then restrict our attention on involutive solutions and consider lambda parametric set-theoretic solutions of the Yang-Baxter equation and we extract the associated quantum algebra. We also discuss the notion of the Drinfeld twist for involutive solutions and their relation to the Yangian. We next focus on reflections and we derive the associated defining algebra relations for R-matrices being Baxterized solutions of the symmetric group. We show that there exists a ``reflection'' finite sub-algebra for some special choice of reflection maps.
Paper Structure (6 sections, 12 theorems, 51 equations)

This paper contains 6 sections, 12 theorems, 51 equations.

Table of Contents

  1. The set-theoretic Yang-Baxter equation
  2. Set-theoretic Yang Baxter equation $\&$ quantum groups
  3. The Quantum Algebra associated to braces
  4. Set-theoretic reflection equation
  5. Reflection algebra from set-theoretic solutions
  6. Conclusions

Key Result

Theorem 1.2

(Rump's theorem, [25][26]). Assume $(B, +, \circ)$ is a left brace. If the map $\check r_B: B\times B \to B \times B$ is defined as ${\check r}_B(x,y)=(\sigma _{x}(y), \tau _{y}(x))$, where $\sigma _{x}(y)=x\circ y-x$, $\tau _{y}(x)=t\circ x-t$, and $t$ is the inverse of $\sigma _{x}(y)$ in the circ

Theorems & Definitions (32)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3: Rump
  • Theorem 1.4: Theorem GV
  • Remark 1.5
  • Lemma 1.6
  • proof
  • Lemma 1.7
  • proof
  • Lemma 1.8
  • ...and 22 more