${\rm K}_1(\mS_1)$与二元多项式代数的单侧逆代数$\mS_2$的自同构群
${\rm K}_1(\mS_1)$ and the group of automorphisms of the algebra $\mS_2$ of one-sided inverses of a polynomial algebra in two variables
摘要：
在特征为零的域上的二元多项式代数的单侧逆的代数$\mS_2$的自同构群$G_2$中找到了显式生成元。本文还证明了$$ G_2\simeq S_2\ltimes \mT^2\ltimes \Z\ltimes ((K^*\ltimes E_\infty (\mS_1))\boxtimes_{\GL_\infty (K)}(K^*\ltimes E_\infty (\mS_1)))$$，其中$S_2$是对称群，$\mT^2$是二维环面，$E_\infty (\mS_1)$是由初等矩阵生成的$\GL_\infty (\mS_1)$的子群.在证明中，我们利用并证明了关于算子指数的几个结果，证明中的最后一个论点是${\rm K}_1 (\mS_1) \simeq K^*$在文中证明的事实。代数$\mS_1$和$\mS_2$是非交换的，非诺特的，不是域。代数$\mS_2$的单位群被发现（它是巨大的）。
英文摘要：
Explicit generators are found for the group $G_2$ of automorphisms of the algebra $\mS_2$ of one-sided inverses of a polynomial algebra in two variables over a field of characteristic zero. Moreover, it is proved that $$ G_2\simeq S_2\ltimes \mT^2\ltimes \Z\ltimes ((K^*\ltimes E_\infty (\mS_1))\boxtimes_{\GL_\infty (K)}(K^*\ltimes E_\infty (\mS_1)))$$ where $S_2$ is the symmetric group, $\mT^2$ is the 2-dimensional torus, $E_\infty (\mS_1)$ is the subgroup of $\GL_\infty (\mS_1)$ generated by the elementary matrices. In the proof, we use and prove several results on the index of operators, and the final argument in the proof is the fact that ${\rm K}_1 (\mS_1) \simeq K^*$ proved in the paper. The algebras $\mS_1$ and $\mS_2$ are noncommutative, non-Noetherian, and not domains. The group of units of the algebra $\mS_2$ is found (it is huge).
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