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

Abstract

a b s t r a c tFor the algebra I 1 = K ⟨x, d dx , ⟩ of polynomial integro-differential operators over a field K of characteristic zero, a classification of simple modules is given. It is proved that I 1 is a left and right coherent algebra. The Strong Compact-Fredholm Alternative is proved for I 1 . The endomorphism algebra of each simple I 1 -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 I 1 exists but there exists the largest left quotient ring and the largest right quotient ring of I 1 , they are not I 1 -isomorphic but I 1 -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 I 1 : each finitely generated one-sided ideal of I 1 is 2-generated.

Polynomial Solutions and Annihilators of Ordinary Integro-Differential Operators

“…Finally, in [1,2,3], various algebraic properties of I and important results are proven amongst them a classification of simple modules, an analogue of Stafford's theorem, and of the first conjecture of Dixmier.…”

“…The algebra I(k), simply be denoted by I in what follows, was studied in [1,3] as a generalized Weyl algebra. See [20] for the construction of I as a factor algebra of a skew polynomial ring.…”

“…In contrast to [1,3], this last approach allows one to have more than one point evaluation, which is crucial for the study of boundary value problems.…”

“…The integral operator is generally eliminated by differentiating once (1) to get the following linear ordinary differential (OD) equation:…”

“…Algebras of ordinary ID operators have recently been studied within an algebraic approach in [1,2,3,4] and within an algorithmic approach in [20,21,22]. The goal of the latter works is to provide an algebraic and algorithmic framework for studying boundary value problems and Green's operators.…”