$GL_n(R)$、$O_{2n}(R)$和$U_{2n}(R,Λ)$的夹层分类
Sandwich classification for $GL_n(R)$, $O_{2n}(R)$ and $U_{2n}(R,Λ)$ revisited
摘要：
设$n$是大于或等于$3$的自然数，$R$是交换环，$σ\in GL_n(R)$是交换环.我们表明，$t_{kl}(σ_{ij})$（分别）。$t_{kl}(σ_{ii}-σ_{jj}))$，其中$i\neq j$和$k\neq l$可以表示为$8$的产物（分别为$24$）形式的矩阵，其中$^εσ^{\pm 1}$.我们证明了类似的结果正交群$O_{2n}(R)$和双曲酉群$U_{2n}(R,Λ)$的假设下，$R$是交换和$n\geq 3$。这产生了新的，非常简短的证明三明治分类定理的群体$GL_n(R)$，$O_{2n}(R)$和$U_{2n}(R,Λ)$。
英文摘要：
Let $n$ be a natural number greater or equal to $3$, $R$ a commutative ring and $σ\in GL_n(R)$. We show that $t_{kl}(σ_{ij})$ (resp. $t_{kl}(σ_{ii}-σ_{jj}))$ where $i\neq j$ and $k\neq l$ can be expressed as a product of $8$ (resp. $24$) matrices of the form $^εσ^{\pm 1}$ where $ε\in E_n(R)$. We prove similar results for the orthogonal groups $O_{2n}(R)$ and the hyperbolic unitary groups $U_{2n}(R,Λ)$ under the assumption that $R$ is commutative and $n\geq 3$. This yields new, very short proofs of the Sandwich Classification Theorems for the groups $GL_n(R)$, $O_{2n}(R)$ and $U_{2n}(R,Λ)$.
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。
