arXiv 1902.009622019-02-03math.QA

杨-巴克斯特方程无平方解的辫群及其群代数

The braided group of a square-free solution of the Yang-Baxter equation and its group algebra

Tatiana Gateva-Ivanova

备注：: 14 pages. arXiv admin note: text overlap with arXiv:1507.02602

摘要：

杨-巴克斯特方程的集合论解是数学物理、代数和组合学的交汇点。这样的解$(X,r)$由一个集合$X$和一个满足辫子关系的双射映射$r:X\times X\to X\times X$组成。本文研究了有限阶$n$和循环指数$p=p(X,r)$的对合无平方解$(X,r)$的辫群$G=G(X,r)$和域$\textbf{k}$上的群代数代数$\textbf{k} [G]$.证明了$G$包含一个有限指标$p^n$的$G$ -不变正规子群$\mathcal{F}_p$，$\mathcal{F}_p$同构于秩为$n$的自由阿贝尔群.我们明确地描述了$p^n$阶的商辫群$\widetilde{G}=G/\mathcal{F}_p$，并证明了$X$嵌入在$\widetilde{G}$中.本文证明了群代数$\textbf{k} [G]$是自由左（分别是右）模，并给出了它的交换子代数$\textbf{k}[\mathcal{F}_p]$上的有限秩$p^n$的显式自由基。$\textbf{k} [G]$的中心包含$\textbf{k} [x_1^p, \cdots, x_n^p]$中对称多项式的子代数。群环上的经典结果暗示$\textbf{k}[G]$是有限整体维数的左（和右）Noether整环。

英文摘要：

Set-theoretic solutions of the Yang--Baxter equation form a meeting-ground of mathematical physics, algebra and combinatorics. Such a solution $(X,r)$ consists of a set $X$ and a bijective map $r:X\times X\to X\times X$ which satisfies the braid relations. In this work we study the braided group $G=G(X,r)$ of an involutive square-free solution $(X,r)$ of finite order $n$ and cyclic index $p=p(X,r)$ and the group algebra $\textbf{k} [G]$ over a field $\textbf{k}$. We show that $G$ contains a $G$-invariant normal subgroup $\mathcal{F}_p$ of finite index $p^n$, $\mathcal{F}_p$ is isomorphic to the free abelian group of rank $n$. We describe explicitly the quotient braided group $\widetilde{G}=G/\mathcal{F}_p$ of order $p^n$ and show that $X$ is embedded in $\widetilde{G}$. We prove that the group algebra $\textbf{k} [G]$ is a free left (resp. right) module of finite rank $p^n$ over its commutative subalgebra $\textbf{k}[\mathcal{F}_p]$ and give an explicit free basis. The center of $\textbf{k} [G]$ contains the subalgebra of symmetric polynomials in $\textbf{k} [x_1^p, \cdots, x_n^p]$. Classical results on group rings imply that $\textbf{k}[G]$ is a left (and right) Noetherian domain of finite global dimension.

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摘要：

英文摘要：

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