$O_{2n+1}(R)$和$U_{2n+1}(R,Δ)$的夹层分类
Sandwich classification for $O_{2n+1}(R)$ and $U_{2n+1}(R,Δ)$ revisited
摘要：
在最近的一篇文章中，作者证明了：如果$n\geq 3$是自然数，$R$是交换环，$σ\in GL_n(R)$是交换环，则$t_{kl}(σ_{ij})$，其中$i\neq j$和$k\neq l$可以表示为形式为$^εσ^{\pm 1}$的$8$矩阵的乘积，其中$ε\in E_n(R)$ .在这篇文章中，我们证明了类似的结果，奇数维正交群$O_{2n+1}(R)$和奇数维酉群$U_{2n+1}(R,Δ)$的假设下，$R$是交换和$n\geq 3$。这产生了新的，简短的证明三明治分类定理的群体$O_{2n+1}(R)$和$U_{2n+1}(R,Δ)$。
英文摘要：
In a recent paper, the author proved that if $n\geq 3$ is a natural number, $R$ a commutative ring and $σ\in GL_n(R)$, then $t_{kl}(σ_{ij})$ where $i\neq j$ and $k\neq l$ can be expressed as a product of $8$ matrices of the form $^εσ^{\pm 1}$ where $ε\in E_n(R)$. In this article we prove similar results for the odd-dimensional orthogonal groups $O_{2n+1}(R)$ and the odd-dimensional unitary groups $U_{2n+1}(R,Δ)$ under the assumption that $R$ is commutative and $n\geq 3$. This yields new, short proofs of the Sandwich Classification Theorems for the groups $O_{2n+1}(R)$ and $U_{2n+1}(R,Δ)$.
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