A combinatorial approach to the set-theoretic solutions of the Yang–Baxter equation
A combinatorial approach to the set-theoretic solutions of the Yang–Baxter equation
复制标题
杨-巴克斯特方程集合论解的组合方法
DOI：
--
复制
发表时间：
2004
期刊：
影响因子：
--
通讯作者：
T. Gateva
中科院分区：
文献类型：
--
作者：
T. Gateva
双射映射r：x2→x2，其中X={x1，…，xn}是有限集，如果辫子关系r12r23r12=r23r12r23在X3中成立，则称为Yang-Baxter方程(YBE)的集合论解。对所有x∈X，满足r(Xx)=xx的非退化对合解(X，r)称为无平方解。YBE的无平方集论解、I型半群、斜多项式型半群和Bieberbach群之间存在着密切的联系，这一点在他与Michel Van den Bergh的一篇联合论文中首次被证明。本文继续研究域k上的无平方解(X，r)及其相关的Yang-Baxter代数结构--域k上的半群S(X，r)、群G(X，r)和k-代数A(k，X，r)，它们是由X生成的，二次定义关系自然地由r唯一确定。我们研究了相关的Yang-Baxter结构的性质，并证明了作者的一个猜想：(集合论)ybe的一个无平方解，…
A bijective map r: X2→X2, where X={x1,…,xn} is a finite set, is called a set-theoretic solution of the Yang–Baxter equation (YBE) if the braid relation r12r23r12=r23r12r23 holds in X3. A nondegenerate involutive solution (X,r) satisfying r(xx)=xx, for all x∈X, is called square-free solution. There exist close relations between the square-free set-theoretic solutions of YBE, the semigroups of I-type, the semigroups of skew polynomial type, and the Bieberbach groups, as it was first shown in a joint paper with Michel Van den Bergh. In this paper we continue the study of square-free solutions (X,r) and the associated Yang–Baxter algebraic structures—the semigroup S(X,r), the group G(X,r) and the k-algebra A(k,X,r) over a field k, generated by X and with quadratic defining relations naturally arising and uniquely determined by r. We study the properties of the associated Yang–Baxter structures, and prove a conjecture of the present author that the three notions: a square-free solution of (set-theoretic) YBE, ...