Commutative subalgebras of the ring of differential operators on a curve

Patrick Thomas Perkins

Vol. 139 (1989), No. 2, 279–302

DOI: 10.2140/pjm.1989.139.279

Abstract

Let X denote an irreducible affine algebraic curve over the complex numbers. Let 𝒪(X) be the ring of regular functions on X. Denote by 𝒟(X) the ring of differential operators on X. We wish to characterize 𝒪(X) as a ring theoretic invariant of 𝒟(X). It is proved that 𝒪(X) equals the set of all locally ad-nilpotent elements of 𝒟(X) if and only if X is not simply connected. However, for most simply connected curves, we show there exists a maximal commutative subalgebra of 𝒟(X), consisting of locally ad-nilpotent elements, which is not isomorphic to 𝒪(X).

Mathematical Subject Classification 2000

Primary: 14H20

Secondary: 17B35, 32C38, 16A72, 16A65

Milestones

Received: 1 March 1988

Published: 1 October 1989