The algebra of integro-differential operators on a polynomial algebra

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 ($\le 2^{2^n}$), 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.

27 pages

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• On Integro-Differential Algebras
• Symbolic Analysis for Boundary Problems: From Rewriting to Parametrized Gröbner Bases
• Composing and Factoring Generalized Green's Operators and Ordinary Boundary Problems
• An analogue of the Conjecture of Dixmier is true for the algebra of polynomial integro-differential operators
• The fundamental theorem of calculus in differential rings
• The group of automorphisms of the algebra of polynomial integro-differential operators
• The algebra of polynomial integro-differential operators is a holonomic bimodule over the subalgebra of polynomial differential operators
• The largest left quotient ring of a ring
• The global dimension of the algebras of polynomial integro-differential operators ${\mathbb{I}}_n$ and the Jacobian algebras ${\mathbb{A}}_n$
• The algebra of integro-differential operators on an affine line and its modules
• Explicit description of generalized weight modules of the algebra of polynomial integro-differential operators $I_n$
• Indecomposable generalized weight modules over the algebra of polynomial integro-differential operators
• Weakly left localizable rings
• Study of the algebra of smooth integro-differential operators with applications