论文标题
关于Yang-baxter方程的各种类型的结构的nil蛋白
On various types of nilpotency of the structure monoid and group of a set-theoretic solution of the Yang--Baxter equation
论文作者
论文摘要
鉴于有限的射线非脱位设定理论解决方案$（x，r）$的杨 - 巴克斯特方程，当它的结构monoid $ m（x，r）$是malcev nilpotent时，我们会表征其表征。将此特征应用于来自架子的解决方案，我们重新发现了Lebed and Mortier最近获得的一些结果，以及Lebed和Vendramin在描述有限的Abelian架子和Quandles的描述中。 我们还调查了五个$（x，r）$的肉类非分类多样性（不一定是有限的）解决方案，例如，表明该属性与与结构$ m（x，r）$（x，r）$（分别结构$ g（x，r）的结构相关的解决方案相同由一个人扩大，概括了在相关情况下获得的Gateva-Ivanova和Cameron的结果。此外，我们还证明，如果$ x $是有限的，而$ g = g（x，r）$是nilpotent的，那么组$ g $的扭转部分是有限的，它与换向量$ g $ g $ g $ g/g/g/g/g $ g $ g/g $的换向器子组$ [g，g] _+$相吻合。
Given a finite bijective non-degenerate set-theoretic solution $(X,r)$ of the Yang--Baxter equation we characterize when its structure monoid $M(X,r)$ is Malcev nilpotent. Applying this characterization to solutions coming from racks, we rediscover some results obtained recently by Lebed and Mortier, and by Lebed and Vendramin on the description of finite abelian racks and quandles. We also investigate bijective non-degenerate multipermutation (not necessarily finite) solutions $(X,r)$ and show, for example, that this property is equivalent to the solution associated to the structure monoid $M(X,r)$ (respectively structure group $G(X,r)$) being a multipermuation solution and that $G=G(X,r)$ is solvable of derived length not exceeding the multipermutation level of $(X,r)$ enlarged by one, generalizing results of Gateva-Ivanova and Cameron obtained in the involutive case. Moreover, we also prove that if $X$ is finite and $G=G(X,r)$ is nilpotent, then the torsion part of the group $G$ is finite, it coincides with the commutator subgroup $[G,G]_+$ of the additive structure of the skew left brace $G$ and $G/[G,G]_+$ is a trivial left brace.
