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

Abstract

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.