代数$B_q(f)$的臭氧群

中文摘要 AI 辅助

设$q$为$n$次本原单位根，$n>1$，$f$为非零多项式，满足对所有$j\in\supp(f)$有$n\nmid(j+1)$，记$e=\gcd(n,\{j+1:j\in\supp(f)\})$。我们证明$\Oz(B_q(f))\cong\mu_e\times\mu_e$：其定义关系对$\mathbb{Z}/e\times\mathbb{Z}/e$分次是齐次的，中心位于零次，臭氧群是该分次的特征群。臭氧群的确定仅需中心元素$u^n$、$v^n$和$\Omega$。模中心的正则正规元素构成由$u^{n/e}$和$v^{n/e}$生成的同一群，故当且仅当$e=1$时所有正规元素都是中心元素。对$f=t^2$，我们得到Chan、Gaddis、Won和Zhang的计算结果，且对$e>1$，我们得到一族具有非平凡臭氧群的无限Calabi–Yau代数。

英文摘要

Let $q$ be a primitive $n$-th root of unity, $n>1$, and let $f$ be a nonzero polynomial such that $n\nmid(j+1)$ for every $j\in\supp(f)$. Set $e=\gcd(n,\{j+1:j\in\supp(f)\})$. We show that $\Oz(B_q(f))\congμ_e\timesμ_e$: the defining relations are homogeneous for a $\mathbb{Z}/e\times\mathbb{Z}/e$ grading, the center sits in degree zero, and the ozone group is the character group of that grading. The determination of the ozone group only requires the central elements $u^n$, $v^n$, and $Ω$. The regular normal elements modulo the center form the same group, generated by $u^{n/e}$ and $v^{n/e}$, so every normal element is central exactly when $e=1$. For $f=t^2$ we recover a computation of Chan, Gaddis, Won and Zhang, and for $e>1$ we obtain an infinite family of Calabi--Yau algebras with nontrivial ozone group.

发表机构

• Universidad Pedagógica y Tecnológica de Colombia（哥伦比亚教育技术大学）
• Universidad Militar Nueva Granada（新格拉纳达军事大学）

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