# Introduction to the Yang-Baxter Equation with Open Problems

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition.**A linear automorphism R of is a solution of the Yang-Baxter equation (sometimes called the braid relation), if the equality

**Definition.**R is a solution of the quantum Yang-Baxter equation (QYBE) if

_{ij}means R acting on the i-th and j-th component.

**Theorem 1.**(D. Hobby and F. F. Nichita, [10]). Assume R is a reflexive relation. The function S derived from the relation R satisfies (3) if and only if R R

^{op}is an equivalence relation on X and the complement relation of R is a strict partial order on each class of R R

^{op}(where R

^{op}is the opposite relation of R).

**Remark.**The above theorem generalizes the twist map if R is an equivalence relation. For a vector space, if we give an equivalence relation on a basis of the space, we can construct a generalization of the twist map using formula (4).

**Remark.**

**[10]**compared the above solutions with the solutions from Boolean algebras. More precisely, the function is a solution for the set-theoretical Yang-Baxter equation.

## 3. Main Results and Discussion

**Remarks.**

**Definition.**Two YB operators (V, R) and (W, Q) are called isomorphic if there exists such that .

**Remark.**Reference [11] showed that for non-isomorphic algebra structures, the associated YB operators are non-isomorphic. It follows that, for any finite dimension vector space, the number of non-isomorphic classes of algebras structures on that vector space is less than the number of non-isomorphic YB operators on the same vector space.

**Conjecture.**The YB operators from algebra structures can be obtained from some kind of universal R-matrix (a universal R-matrix is related to the quasi-triangular structures presented in [6]).

**Theorem 2.**There exist finite dimensional vector spaces for which there are infinitely many non-isomorphic Yang-Baxter operators.

**Proof.**The idea of the proof is shown below. The omitted technical details will be included in another paper. We take an arbitrary Hopf algebra from [17]. (We know that there are infinitely many non-isomorphic finite dimensional Hopf algebras.) Because this Hopf algebra can be viewed as an entwining structure, we associate a WXZ system as in [12]. But X is invertible, since its inverse could be obtained by using the antipode of this arbitrary Hopf algebra. We now construct a new Yang-Baxter operator using a remark from [12], because X is invertible:

## 4. Conclusions and Directions for Future Research

## Acknowledgments

## References and Notes

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Nichita, F.
Introduction to the Yang-Baxter Equation with Open Problems. *Axioms* **2012**, *1*, 33-37.
https://doi.org/10.3390/axioms1010033

**AMA Style**

Nichita F.
Introduction to the Yang-Baxter Equation with Open Problems. *Axioms*. 2012; 1(1):33-37.
https://doi.org/10.3390/axioms1010033

**Chicago/Turabian Style**

Nichita, Florin.
2012. "Introduction to the Yang-Baxter Equation with Open Problems" *Axioms* 1, no. 1: 33-37.
https://doi.org/10.3390/axioms1010033