论文标题
将有限的矩阵表达到有限订单元素的换向器的产品中
Expressing Finite-Infinite Matrices Into Products of Commutators of Finite Order Elements
论文作者
论文摘要
令$ r $为一个unity $ 1 $的关联戒指，并考虑$ k \ in \ mathbb {n} $，这样$ 1+1+1+..+1 = k $是可逆的。用$ω$表示$ r $的任意kth unity的统一根，让$ ut^{（k）} _ {\ infty}（r）$是上层三角无限矩阵的组，其对角线的对角为$ k $ th的根源为$ 1 $。我们表明，组的每个元素$ ut _ {\ infty}（r）$可以表示为$ 4K-6 $换向器的产品，这些产品取决于$ ut^{（k）} _ {\ infty} _ {\ infty}（r）$ k $的元素元素的元素。如果$ r $是复杂的字段或实际数字字段，我们证明，在$ sl_n（r）$中，在子组$ sl_ {vk}（\ infty，r）$ y $ r $的$ r $中，这些组中的每个元素都可以分解为最多4k-6 $ $ k $ $ k $ $ k $ $ k $的$ 4K-6 $通勤者的产品。
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 $k$.
