量子 Yang-Baxter 方程的“集合论”解与一类 Garside 群

群、预群、括号、Yang-Baxter方程和$T$-结构上权为0的相对Rota-Baxter算子

Relative Rota-Baxter operators of weight 0 on groups, pre-groups, braces, the Yang-Baxter equation and $T$-structures

本文研究群和李群上权为0的相对Rota-Baxter算子，并建立了其与预群、括号、Yang-Baxter方程集合论解及$T$-结构之间的显式关系。

关于 Garside 群中的左序

About left orders in Garside groups

研究量子 Yang-Baxter 方程解的结构群，证明其不可双序，并揭示其左不变全序存在性的多样性。

Abelian fixed point free endomorphisms and the Yang-Baxter equation

We obtain a simple family of solutions to the set-theoretic Yang-Baxter equation, one which depends only on considering special endomorphisms of a finite group. We show how such an endomorphism gives rise to two non-degenerate solutions to the Yang-Baxter equation, solutions which are inverse to each other. We give concrete examples using dihedral, alternating, symmetric, and metacyclic groups.

Dehornoy类与集合论杨-巴克斯特方程解的Sylow子群

Dehornoy's class and Sylows for set-theoretical solutions of the Yang-Baxter equation

本文研究循环集的结构群分解为Sylow子群，探讨Dehornoy类并猜想界，结合括号与单项式表示回答Dehornoy问题并恢复Rump定理。

Coxeter-like groups for groups of set-theoretic solutions of the Yang--Baxter equation

We attach with every finite, involutive, nondegenerate set-theoretic solution of the Yang--Baxter equation a finite group that plays for the associated structure group the role that a finite Coxeter group plays for the associated Artin--Tits group.

On groups of $I$-type and involutive Yang-Baxter groups

We suggest a cohomological framework to describe groups of $I$-type and involutive Yang-Baxter groups. These groups are key in the study of involutive non-degenerate set-theoretic solutions of the quantum Yang-Baxter equation. Our main tool is a lifting criterion for 1-cocycles, established here in a general non-abelian setting.

Braces and the Yang-Baxter equation

Several aspects of relations between braces and non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation are discussed and many consequences are derived. In particular, for each positive integer $n$ a finite square-free multipermutation solution of the Yang-Baxter equation with multipermutation level $n$ and an abelian involutive Yang-Baxter group is constructed. This answers a problem of Gateva-Ivanova and Cameron. It is also proved that finite non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation whose associated involutive Yang-Baxter group is ab…

Indecomposable involutive solutions of the Yang-Baxter equation of multipermutation level 2 with non-abelian permutation group

We give a complete characterization of all indecomposable involutive solutions of the Yang-Baxter equation of multipermutation level~2. In the first step we present a construction of some family of such solutions and in the second step we prove that every indecomposable involutive solution of the Yang-Baxter equation with multipermutation level~2 is a homomorphic image of a solution previously constructed. Analyzing this epimorphism, we are able to obtain all such solutions up to isomorphism and enumerate these of small sizes.

与Yang-Baxter方程的有限非退化解相关联的结构斜撑是有限表示的

The structure skew brace associated with a finite non-degenerate solution of the Yang-Baxter equation is finitely presented

本文证明了与Yang-Baxter方程的有限非退化解相关联的结构斜撑具有有限表示性质。

A characterization of finite multipermutation solutions of the Yang-Baxter equation

We prove that a finite non-degenerate involutive set-theoretic solution (X,r) of the Yang-Baxter equation is a multipermutation solution if and only if its structure group G(X,r) admits a left ordering or equivalently it is poly-(infinite cyclic).

Drinfeld-Manin solutions of the Yang-Baxter equation coming from cube complexes

The most common geometric interpretation of the Yang-Baxter equation is by braids, knots and relevant Reidemeister moves. So far, cubes were used for connections with the third Reidemeister move only. We will show that there are higher-dimensional cube complexes solving the $D$-state Yang-Baxter equation for arbitrarily large $D$. More precisely, we introduce explicit constructions of cube complexes covered by products of $n$ trees and show that these cube complexes lead to new solutions of the Yang-Baxter equations.