Clifford代数的等距I
Isometries of Clifford Algebras I
摘要：
设$V$是特征不等于2的域$F$上的有限维向量空间，$Q$是$V$上的非退化对称双线性型.设$C\ell(V,Q)$是由$V$和$Q$确定的Clifford代数.双线性形式$Q$以自然的方式扩展到$C\ell(V,Q)$上的非退化、对称、双线性形式$\bar{Q}$。设$G$是$C\ell(V,Q)$相对于$\bar{Q}$的等距群，$LG$是$C\ell(V,Q)$相对于$\bar{Q}$的无穷小等距的李代数.我们导出了$LG$的一些基本结构信息，并在$F = R, V = R^{n}$和$Q$在$R^{n}$上正定的情况下计算了$G$.在本文的续篇中，我们在$F = R, V = R^{n}$和$Q$在$R^{n}$上非退化的情况下，确定了$LG$.
英文摘要：
Let $V$ be a finite dimensional vector space over a field $F$ of characteristic different from 2, and let $Q$ be a nondegenerate, symmetric, bilinear form on $V$. Let $C\ell(V,Q)$ be the Clifford algebra determined by $V$ and $Q$. The bilinear form $Q$ extends in a natural way to a nondegenerate, symmetric, bilinear form $\bar{Q}$ on $C\ell(V,Q)$. Let $G$ be the group of isometries of $C\ell(V,Q)$ relative to $\bar{Q}$, and let $LG$ be the Lie algebra of infinitesimal isometries of $C\ell(V,Q)$ relative to $\bar{Q}$. We derive some basic structural information about $LG$, and we compute $G$ in the case that $F = R, V = R^{n}$ and $Q$ is positive definite on $R^{n}$. In a sequel to this paper we determine $LG$ in the case that $F = R, V = R^{n}$ and $Q$ is nondegenerate on $R^{n}$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
