Algebra Of Integro-differential Operators Research Articles

The global dimension of the algebras of polynomial integro-differential operators 𝕀n and the Jacobian algebras 𝔸n

The aim of the paper is to prove two conjectures from the paper [V. V. Bavula, The algebra of integro-differential operators on a polynomial algebra, J. London Math. Soc. (2) 83 (2011) 517–543, arXiv:math.RA/0912.0723] that the (left and right) global dimension of the algebra [Formula: see text] of polynomial integro-differential operators and the Jacobian algebra [Formula: see text] is equal to [Formula: see text] (over a field of characteristic zero). The algebras [Formula: see text] and [Formula: see text] are neither left nor right Noetherian and [Formula: see text]. Furthermore, they contain infinite direct sums of nonzero left/right ideals and are not domains. An analogue of Hilbert’s Syzygy Theorem is proven for the algebras [Formula: see text], [Formula: see text] and their factor algebras. It is proven that the global dimension of all prime factor algebras of the algebras [Formula: see text] and [Formula: see text] is [Formula: see text] and the weak global dimension of all the factor algebras of [Formula: see text] and [Formula: see text] is [Formula: see text].

Study of the algebra of smooth integro-differential operators with applications

We study the algebra of integro-differential operators with smooth coefficients and kernels on a subspace of [Formula: see text]. We find a normal form for elements of this algebra and determine its unit group. The formulation of inverses gives explicit solutions of inhomogeneous linear Volterra integro-differential equations and Volterra integral equations of first kind with smooth kernels.

The algebra of integro-differential operators on an affine line and its modules

For the algebra I1=K〈x,ddx,∫〉 of polynomial integro-differential operators over a field K of characteristic zero, a classification of simple modules is given. It is proved that I1 is a left and right coherent algebra. The Strong Compact-Fredholm Alternative is proved for I1. The endomorphism algebra of each simple I1-module is a finite dimensional skew field. In contrast to the first Weyl algebra, the centralizer of a nonscalar integro-differential operator can be a noncommutative, non-Noetherian, non-finitely generated algebra which is not a domain. It is proved that neither left nor right quotient ring of I1 exists but there exists the largest left quotient ring and the largest right quotient ring of I1, they are not I1-isomorphic but I1-anti-isomorphic. Moreover, the factor ring of the largest right quotient ring modulo its only proper ideal is isomorphic to the quotient ring of the first Weyl algebra. An analogue of the Theorem of Stafford (for the Weyl algebras) is proved for I1: each finitely generated one-sided ideal of I1 is 2-generated.

The algebra of integro-differential operators on a polynomial algebra

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.