${\rm K}_1(\mS_1)$ and the group of automorphisms of the algebra $\mS_2$ of one-sided inverses of a polynomial algebra in two variables

Abstract

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).