The group of automorphisms of the algebra of one-sided inverses of a polynomial algebra

The algebra ${\text{mS}}_n$ in the title is obtained from a polynomial algebra $P_n$ in $n$ variables by adding commuting, {\em left} (but not two-sided) inverses of the canonical generators of $P_n$. Ignoring non-Noetherian property, the algebra ${\text{mS}}_n$ belongs to a family of algebras like the Weyl algebra $A_n$ and the polynomial algebra $P_{2n}$. The group of automorphisms $G_n$ of the algebra ${\text{mS}}_n$ is found: $$G_n=S_n⋉{\text{mT}}^n⋉\text{Inn}({\text{mS}}_n)\supseteq S_n⋉{\text{mT}}^n⋉\underset{2^n-1\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{s}}{\underbrace{{\text{GL}}_{\mathrm{\infty }}(K)⋉\mathrm{.}\mathrm{.}\mathrm{.}⋉{\text{GL}}_{\mathrm{\infty }}(K)}}=:G_n^{\mathrm{\prime }}$$ where $S_n$ is the symmetric group, ${\text{mT}}^n$ is the $n$-dimensional torus, $\text{Inn}({\text{mS}}_n)$ is the group of inner automorphisms of ${\text{mS}}_n$ (which is huge), and ${\text{GL}}_{\mathrm{\infty }}(K)$ is the group of invertible infinite dimensional matrices. This result may help in understanding of the structure of the groups of automorphisms of the Weyl algebra $A_n$ and the polynomial algebra $P_{2n}$. An analog of the {\em Jacobian homomorphism}: ${\text{Aut}}_{K-\mathrm{a}\mathrm{l}\mathrm{g}}(P_{2n})\text{ra}{K}^{\ast }$, so-called, the {\em global determinant} is introduced for the group $G_n^{\mathrm{\prime }}$ (notice that the algebra ${\text{mS}}_n$ is {\em noncommutative} and neither left nor right Noetherian).