The algebra of polynomial integro-differential operators is a holonomic bimodule over the subalgebra of polynomial differential operators

TL;DR

This paper studies the algebra of polynomial integro-differential operators, revealing its structure as a holonomic bimodule over polynomial differential operators, and provides new canonical forms and properties related to its socle filtration.

Contribution

It establishes that the algebra of polynomial integro-differential operators is a holonomic bimodule with specific socle structure, and introduces a new canonical form for these operators.

Findings

The algebra is a holonomic $A_n$-bimodule of length $3^n$.

The socle length of the bimodule is $n+1$, with a detailed filtration.

The algebra is the maximal order in its quotient ring.

Abstract

In contrast to its subalgebra $A_n:=K<x_1, ..., x_n, \frac{\der}{\der x_1}, ...,\frac{\der}{\der x_n}>$ of polynomial differential operators (i.e. the $n$'th Weyl algebra), 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 neither left nor right Noetherian algebra; moreover it contains infinite direct sums of nonzero left and right ideals. It is proved that $\mI_n$ is a left (right) coherent algebra iff $n=1$; the algebra $\mI_n$ is a {\em holonomic $A_n$-bimodule} of length $3^n$ and has multiplicity $3^n$, and all $3^n$ simple factors of $\mI_n$ are pairwise non-isomorphic $A_n$-bimodules. The socle length of the $A_n$-bimodule $\mI_n$ is $n+1$, the socle filtration is found, and the $m$'th term of the socle filtration has length ${n\choose m}2^{n-m}$. This fact gives a new canonical form for each polynomial integro-differential operator. It is proved that the algebra $\mI_n$ is the maximal left (resp. right) order in the largest left (resp. right) quotient ring of the algebra $\mI_n$.