分次Clifford代数和分次斜Clifford代数及其在Artin-Schelter正则代数分类中的作用
Graded Clifford Algebras and Graded Skew Clifford Algebras and Their Role in the Classification of Artin-Schelter Regular Algebras
摘要：
本文综述了$\mathbb{N}$ -分次Clifford代数（GCA）和$\mathbb{N}$ -分次\textit{skew}-Clifford代数（GSCA）\cite{VVW, SV, CaV, NVZ, VVe1, VVe2}的研究工作.特别是，我们讨论了必要的假设，这些代数是阿廷Schelter定期\cite{AS, ATV1}，并显示如何某些“点”称为，点模块，可以与他们。我们可以把AS-正则代数看作多项式环的非交换模拟。我们从\cite{VVW}中的一个基本结果开始我们的调查，这个结果对这里讨论的后续结果是必不可少的：点模和秩二次曲面之间的连接。部分使用这种连接，作者在\cite{SV}提供了一种方法来构建GCA与许多不同的同构类的点模。在\cite{CaV}中，Cassidy和Vancliff引入了GCA的一个量子化类似物，称为分次\textit{skew} Clifford代数，Nafari等人的\cite{NVZ}证明了大部分整体维数为3的Artin Schelter正则代数要么是整体维数为3的分次斜Clifford代数的twists，要么是整体维数为2的分次Clifford代数的Ore扩张。Vancliff等人的\cite{VVe1, VVe2}更进一步，在点模和秩二次曲面之间推广了\cite{VVW}的结果，证明了GSCA上的点模由$μ$ -秩至多为2的（非交换）二次曲面确定。
英文摘要：
This paper is a survey of work done on $\mathbb{N}$-graded Clifford algebras (GCAs) and $\mathbb{N}$-graded \textit{skew} Clifford algebras (GSCAs) \cite{VVW, SV, CaV, NVZ, VVe1, VVe2}. In particular, we discuss the hypotheses necessary for these algebras to be Artin Schelter-regular \cite{AS, ATV1} and show how certain `points' called, point modules, can be associated to them. We may view an AS-regular algebra as a noncommutative analog of the polynomial ring. We begin our survey with a fundamental result in \cite{VVW} that is essential to subsequent results discussed here: the connection between point modules and rank-two quadrics. Using, in part, this connection the authors in \cite{SV} provide a method to construct GCAs with finitely many distinct isomorphism classes of point modules. In \cite{CaV}, Cassidy and Vancliff introduce a quantized analog of a GCA, called a graded \textit{skew} Clifford algebra and Nafari et al. \cite{NVZ} show that most Artin Schelter-regular algebras of global dimension three are either twists of graded skew Clifford algebras of global dimension three or Ore extensions of graded Clifford algebras of global dimension two. Vancliff et al. \cite{VVe1, VVe2} go a step further and generalize the result of \cite{VVW}, between point modules and rank-two quadrics, by showing that point modules over GSCAs are determined by (noncommutative) quadrics of $μ$-rank at most two.
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