Yang-Baxter方程集论解的结构单群

Pub Date : 2019-12-20 DOI:10.5565/PUBLMAT6522104
F. Cedó, E. Jespers, C. Verwimp
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引用次数: 11

摘要

给定Yang-Baxter方程的一个集论解$(X,r)$,我们用$M=M(X,r。结果表明,在$a$上存在一个$M$的左作用和在$a'$上存在的$M$右作用,并且关于这些作用,分别存在系数为$a$和$a'$的$\pi$和$\pi'$的1-共循环。我们研究了1-并环何时是内射的、满射的或双射的。在$X$是有限的情况下,证明$\pi$是双射的当且仅当$(X,r)$是左非退化的,并且$\pi'$是双反的当且唯当$(X,r)美元是右非退化的。在$(X,r)$是非退化的,特别是$\pi$是双射的情况下,我们定义了$M(X,r)$上的半桁架结构,然后我们证明了这在$M(X,r)美元的最小可消去图像$\bar M=M(X、r)/\eta$上自然地诱导了一个集理论解$(\bar M,\bar r)$。如果$X$自然嵌入$M(X,r)/\eta$中,例如当$(X,r)$不可处理时,则$\bar r$是$r$的扩展。还证明了非退化不可调和解必然是双射的。
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Structure monoids of set-theoretic solutions of the Yang-Baxter equation
Given a set-theoretic solution $(X,r)$ of the Yang--Baxter equation, we denote by $M=M(X,r)$ the structure monoid and by $A=A(X,r)$, respectively $A'=A'(X,r)$, the left, respectively right, derived structure monoid of $(X,r)$. It is shown that there exist a left action of $M$ on $A$ and a right action of $M$ on $A'$ and 1-cocycles $\pi$ and $\pi'$ of $M$ with coefficients in $A$ and in $A'$ with respect to these actions respectively. We investigate when the 1-cocycles are injective, surjective or bijective. In case $X$ is finite, it turns out that $\pi$ is bijective if and only if $(X,r)$ is left non-degenerate, and $\pi'$ is bijective if and only if $(X,r)$ is right non-degenerate. In case $(X,r) $ is left non-degenerate, in particular $\pi$ is bijective, we define a semi-truss structure on $M(X,r)$ and then we show that this naturally induces a set-theoretic solution $(\bar M, \bar r)$ on the least cancellative image $\bar M= M(X,r)/\eta$ of $M(X,r)$. In case $X$ is naturally embedded in $M(X,r)/\eta$, for example when $(X,r)$ is irretractable, then $\bar r$ is an extension of $r$. It also is shown that non-degenerate irretractable solutions necessarily are bijective.
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