发表机构
UC San Diego（加州大学圣迭戈分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文完全分类了三维 Artin--Schelter 正则代数的非交换对偶反射群，发现三次代数无此类群，二次代数中来自 Ore 扩张或 Sklyanin 代数，并构造了新无限族，回答了相关猜想。
AI 中文摘要
根据定义，对偶反射群是一个有限群，它给 Artin--Schelter 正则代数 $A$ 赋予分次结构，使得单位分量 $A_e$ 再次成为正则代数。在本文中，我们完全分类了维数为 $3$ 的正则代数的非交换对偶反射群：我们首先分类所有可能的、细化代数自然 $\mathbb{N}$-分次的非交换分次，然后确定这些分次中哪些产生对偶反射群。我们证明三次代数均不允许对偶反射群，而对于二次代数，对偶反射群要么来自二维正则代数的 Ore 扩张，要么来自 Sklyanin 代数 $S_{q,0,1}$。在 Sklyanin 情形中，对偶反射群构成一个新的无限群族 $\Delta_n$，其阶为 $27n^3$，回答了 Goetz、Kirkman、Moore 和 Vashaw 的一个问题。我们还将这些作者的四维例子推广到两个新的对偶反射群无限族。最后，我们证明允许对偶反射群的 AS 正则代数不必具有二项式关系，回答了他们的另一个问题。然而，我们猜想当 $A_e$ 不含次数为 $1$ 的元素时，关系必须是二项式的，并且我们在维数 $3$ 的非交换分次情形下证明了这一点。
英文摘要
By definition, a dual reflection group is a finite group which grades an Artin--Schelter regular algebra $A$ such that the identity component $A_e$ is again regular. In this paper, we completely classify the nonabelian dual reflection groups for regular algebras of dimension $3$: we first classify all possible nonabelian gradings which refine the natural $\mathbb{N}$-grading of the algebra, and then determine which of these gradings yield dual reflection groups. We show that none of the cubic algebras admit dual reflection groups, while for quadratic algebras, the dual reflection groups arise either from Ore extensions of two-dimensional regular algebras, or from the Sklyanin algebras $S_{q,0,1}$. In the Sklyanin case, the dual reflection groups form a new infinite family of groups $Δ_n$ of order $27n^3$, answering a question of Goetz, Kirkman, Moore, and Vashaw. We also extend the four-dimensional examples of the same authors to two new infinite families of dual reflection groups. Finally, we show that an AS regular algebra admitting a dual reflection group need not have binomial relations, answering another of their questions. However, we conjecture that the relations must be binomial when $A_e$ contains no elements of degree $1$, and we prove this for nonabelian gradings in dimension $3$.
Comments47 pages. Comments welcome!
