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

V. V. Bavula

Published 2009-03-17, updated 2009-06-15Version 3

The algebra $\mS_n$ 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 $\mS_n$ 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 $\mS_n$ is found: $$ G_n=S_n\ltimes \mT^n\ltimes \Inn (\mS_n) \supseteq S_n\ltimes \mT^n\ltimes \underbrace{\GL_\infty (K)\ltimes... \ltimes \GL_\infty (K)}_{2^n-1 {\rm times}}=:G_n' $$ where $S_n$ is the symmetric group, $\mT^n$ is the $n$-dimensional torus, $\Inn (\mS_n)$ is the group of inner automorphisms of $\mS_n$ (which is huge), and $\GL_\infty (K)$ 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}: $\Aut_{K-{\rm alg}}(P_{2n})\ra K^*$, so-called, the {\em global determinant} is introduced for the group $G_n'$ (notice that the algebra $\mS_n$ is {\em noncommutative} and neither left nor right Noetherian).