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arXiv 2004.09012math.RA
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用有穷级元交换子的乘积表示无穷矩阵

Expressing Finite-Infinite Matrices Into Products of Commutators of Finite Order Elements

Ivan Gargate, Michael Gargate

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10 pages
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摘要：

设$R$是具有单位$1$的结合环，并考虑$k\in \mathbb{N}$使得$1+1+..+1=k$可逆.用$ω$表示$R$中的任意k次单位根，设$UT^{(k)}_{\infty}(R)$为对角元为$1$的$k$次根的上三角无限矩阵群.证明了群$UT_{\infty}(R)$中的每个元素都可以表示为$4k-6$算子的乘积，它们都依赖于$k$阶的$UT^{(k)}_{\infty}(R)$中元素的幂.如果$R$是复数域或实数域，我们证明了在$R$上的Vershik-Kerov群的子群$SL_{VK}(\infty,R)$和$SL_n(R)$中，这些群中的每个元素至多可以分解为$k$阶元素的$4k-6$乘子的乘积.

英文摘要：

Let $R$ be an associative ring with unity $1$ and consider $k\in \mathbb{N}$ such that $1+1+..+1=k$ is invertible. Denote by $ω$ an arbitrary kth root of unity in $R$ and let $UT^{(k)}_{\infty}(R)$ be the group of upper triangular infinite matrices whose diagonal entries are $k$th roots of $1$. We show that every element of the group $UT_{\infty}(R)$ can be expressed as a product of $4k-6$ commutators all depending of powers of elements in $UT^{(k)}_{\infty}(R)$ of order $k$. If $R$ is the complex field or the real number field we prove that, in $SL_n(R)$ and in the subgroup $SL_{VK}(\infty,R)$ of the Vershik-Kerov group over $R$, each element in these groups can be decomposed into a product of at most $4k-6$ commutators of elements of order $k$.

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