On the Yang-Baxter equation and left nilpotent left braces
On the Yang-Baxter equation and left nilpotent left braces
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关于 Yang-Baxter 方程和左幂零左括号
DOI：
10.1016/j.jpaa.2016.07.014
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
A. Smoktunowicz
中科院分区：
文献类型：
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作者：
F. Ced'o;T. Gateva;A. Smoktunowicz
研究了Yang-Baxter方程的非退化对合集理论解（X，r），称之为解.证明了有限非平凡解（X，r）的结构群G（X，r）不可能是Engel群.已知有限多置换解（X，r）的结构群G（X，r）是poly-Z群，从而我们的结果给出了辫子群和左括号G（X，r）是poly-Z群而不是Engel群的丰富例子.我们发现了Rump引入的根链A（n+ 1）= A（n）<$A的长度与左支撑的多重置换水平之间的显式关系。我们还证明了杨-巴克斯特方程的有限解可以以一种方便的方式嵌入到有限左支撑中，或者等价地嵌入到有限对合辫群中。
We study non-degenerate involutive set-theoretic solutions (X, r) of the Yang–Baxter equation, we call them solutions. We prove that the structure group G (X, r) of a finite non-trivial solution (X, r) cannot be an Engel group. It is known that the structure group G (X, r) of a finite multipermutation solution (X, r) is a poly-Z group, thus our result gives a rich source of examples of braided groups and left braces G (X, r) which are poly-Z groups but not Engel groups. We find an explicit relation between the multipermutation level of a left brace and the length of the radical chain A (n+ 1)= A (n)⁎ A introduced by Rump. We also show that a finite solution of the Yang–Baxter equation can be embedded in a convenient way into a finite left brace, or equivalently into a finite involutive braided group.