Yang-Baxter置换群作用在分配Yang-Baxter代数上

中文摘要 AI 辅助

设$(X,r)$为Yang-Baxter方程的一个分配集论解，$\u003cbr\u003e\mathcal{A}(X,r)$为相应的Yang-Baxter代数。我们证明$\mathcal{A}(X,r)$同构于一个斜多项式代数，并显式计算其Nakayama自同构。我们研究置换群$\mathcal{G}(X,r)$的作用。它诱导一个子群$\overline{\mathcal{G}}\subseteq\operatorname{Aut}(\mathcal{A}(X,r))$，即诱导自同构群，且$\mathcal{A}(X,r)$是其忠实模。我们刻画了$\overline{\mathcal{G}}$何时为反射群，并在该情形下描述不变子代数$\mathcal{A}(X,r)^{\overline{\mathcal{G}}}$及其Jacobian、反射排列和判别式。我们进一步对一大类分配Yang-Baxter代数建立了Auslander定理。最后，对于一类非平凡的分配Yang-Baxter代数，我们得到了由置换群诱导的群作用的pertinency的一个下界。

英文摘要

Let $(X,r)$ be a distributive set-theoretical solution of the Yang-Baxter equation and $\mathcal{A}(X,r)$ the associated Yang-Baxter algebra. We prove that $\mathcal{A}(X,r)$ is isomorphic to a skew polynomial algebra and compute its Nakayama automorphism explicitly. We study the action of the permutation group $\mathcal{G}(X,r)$. It induces a subgroup $\overline{\mathcal{G}}\subseteq\operatorname{Aut}(\mathcal{A}(X,r))$, the induced automorphism group, for which $\mathcal{A}(X,r)$ is a faithful module. We characterize when $\overline{\mathcal{G}}$ is a reflection group and, in that case, describe the invariant subalgebra $\mathcal{A}(X,r)^{\overline{\mathcal{G}}}$ together with its Jacobian, reflection arrangement and discriminant. We further establish the Auslander theorem for a large class of distributive Yang-Baxter algebras. Finally, for a class of nontrivial distributive Yang--Baxter algebras, we obtain a lower bound for the pertinency of the group action induced by the permutation group.

发表机构

• School of Mathematics, Hangzhou Normal University（杭州师范大学数学学院）

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