The algebra of integro-differential operators on a polynomial algebra

TL;DR

This paper thoroughly analyzes the algebra of integro-differential operators on polynomial algebras, revealing its structural properties, ideal structure, and connections to other algebraic frameworks, including a generalized Weyl algebra and Jacobian algebra.

Contribution

It introduces a detailed structural study of the algebra alI_n, including ideal classification, homological dimensions, and a canonical form, extending known results to a new class of integro-differential operator algebras.

Findings

The algebra alI_n is prime, central, catenary, self-dual, non-Noetherian, with Krull dimension n and Gelfand-Kirillov dimension 2n.

All ideals of alI_n are explicitly described, finite in number, commute, and are idempotent, with their count given by Dedekind numbers.

An analogue of Hilbert's Syzygy Theorem is established for alI_n, and the algebra's units and canonical forms are characterized.

Abstract

We prove that the algebra $\mI_n:=K\langle x_1, ..., x_n, \frac{\der}{\der x_1},...,\frac{\der}{\der x_n}, \int_1, ..., \int_n\rangle $ of integro-differential operators on a polynomial algebra is a prime, central, catenary, self-dual, non-Noetherian algebra of classical Krull dimension $n$ and of Gelfand-Kirillov dimension $2n$. Its weak homological dimension is $n$, and $n\leq \gldim (\mI_n)\leq 2n$. All the ideals of $\mI_n$ are found explicitly, there are only finitely many of them ($\leq 2^{2^n}$), they commute ($\ga \gb = \gb\ga$) and are idempotent ideals ($\ga^2= \ga$). The number of ideals of $\mI_n$ is equal to the {\em Dedekind number} $\gd_n$. An analogue of Hilbert's Syzygy Theorem is proved for $\mI_n$. The group of units of the algebra $\mI_n$ is described (it is a huge group). A canonical form is found for each integro-differential operators (by proving that the algebra $\mI_n$ is a generalized Weyl algebra). All the mentioned results hold for the Jacobian algebra $\mA_n$ (but $\GK (\mA_n) =3n$, note that $\mI_n\subset \mA_n$). It is proved that the algebras $\mI_n$ and $\mA_n$ are ideal equivalent.