The group of automorphisms of the algebra of polynomial integro-differential operators

TL;DR

This paper characterizes the automorphism group of the algebra of polynomial integro-differential operators, revealing its structure, invariants, and providing explicit formulas for automorphism inverses.

Contribution

It explicitly determines the automorphism group of the algebra, describes its structure, invariants, and provides inversion formulas, advancing understanding of polynomial integro-differential operator symmetries.

Findings

The automorphism group al G_n is explicitly described.

Automorphisms are uniquely determined by their action on generators.

The group has trivial center and a finite number of invariant ideals.

Abstract

The group $\rG_n$ of automorphisms of the algebra $\mI_n:=K< x_1, >..., x_n, \frac{\der}{\der x_1}, ... ,\frac{\der}{\der x_n}, \int_1, >..., \int_n>$ of polynomial integro-differential operators is found: $$ \rG_n=S_n\ltimes \mT^n\ltimes \Inn (\mI_n) \supseteq S_n\ltimes \mT^n \ltimes \underbrace{\GL_\infty (K)\ltimes... \ltimes \GL_\infty (K)}_{2^n-1 {\rm times}}, $$ $$ \rG_1\simeq \mT^1 \ltimes \GL_\infty (K),$$ where $S_n$ is the symmetric group, $\mT^n$ is the $n$-dimensional torus, $\Inn (\mI_n)$ is the group of inner automorphisms of $\mI_n$ (which is huge). It is proved that each automorphism $\s \in \rG_n$ is uniquely determined by the elements $\s (x_i)$'s or $\s (\frac{\der}{\der x_i})$'s or $\s (\int_i)$'s. The stabilizers in $\rG_n$ of all the ideals of $\mI_n$ are found, they are subgroups of {\em finite} index in $\rG_n$. It is shown that the group $\rG_n$ has trivial centre, $\mI_n^{\rG_n}=K$ and $\mI_n^{\Inn (\mI_n)}=K$, the (unique) maximal ideal of $\mI_n$ is the {\em only} nonzero prime $\rG_n$-invariant ideal of $\mI_n$, and there are precisely $n+2$ $\rG_n$-invariant ideals of $\mI_n$. For each automorphism $\s \in \rG_n$, an {\em explicit inversion formula} is given via the elements $\s (\frac{\der}{\der x_i})$ and $\s (\int_i)$.