Commutative subalgebras of the ring of differential operators on a curve

Abstract

Let X denote an irreducible affine algebraic curve over the complex numbers. Let &{X) be the ring of regular functions on X. Denote by 3{X) the ring of differential operators on X. We wish to characterize (f(X) as a ring theoretic invariant of 3{X). It is proved that @{X) equals the set of all locally ad-nilpotent elements oi3f(X) if and only if X is not simply connected. However, for most simply connected curves, we show there exists a maximal commutative subalgebra oi3f{X)t consisting of locally ad-nilpotent elements, which is not isomorphic to (f(X). Introduction.Let X be a curve, that is, an irreducible affine algebraic curve over C. Write &{X) for the ring of regular functions on X and 3{X) for the ring of differential operators on X. See [8] for the basic definitions and facts about rings of differential operators on curves. This paper is motivated by the following question. If X and Y are curves with 3{X) = 3(Y) 9 is X = YΊ Write X for the normalization of X. Stafford [9] considers this question for X with X = A 1 , the affine line. He shows that 3{X) = 2{X) if and only if X = X. He also shows that if X is the cubic cusp y 2 = x 3 and Ϋ = A 1 , then X = Y if and only if 3{X) = 3(Y). Higher dimensional non-isomorphic varieties can have isomorphic rings of differential operators, see Levasseur, Smith and Stafford [2].If u e 3{X\ define ad(w) e End c (^(X)) by ad(κ)(u) = [u,v] = uυ -vu. We say u is locally ad-nilpotent if for every v e 2f{X) there exists neN with ad(u) n (v) = 0. Write jr(X) = {ue 3J{X)\u is locally ad-nilpotent}.Note that if ϋ: 3{Y) -> 3{X) is an isomorphism then ϋ(yT(Y)) = Jf(X). It follows from the definition oΐ3{X) that &{X) is a maximal commutative subalgebra oϊ3J(X) and that @(X) is contained in yV{X). If genus {X) > 0 then shows that &{X) = jr\x). Hence if3(X) = 3(Y) with genus(X) > 0 then X = Y.This paper expands on Makar-Limanov's result to prove the following theorem. Let π: X -> X denote the canonical surjection.

Mad subalgebras of rings of differential operators on curves

“…Clearly, the ring O(X) of regular functions on X is a mad subalgebra of D(X). Theorems of Makar-Limanov and Perkins (see [17,18]) show that O(X) is the only mad subalgebra except in the case when X is a framed curve, by which we mean that there is a regular bijective map π : A 1 → X (thus topologically a framed curve is simply the affine line, but it may have an arbitrary finite number of cusps). From now on we suppose that X is a framed curve, since Theorems 1.1 and 1.2 can have interesting generalizations only in that case.…”

“…The new part of Theorem 1.3 is thus the assertion that multiple points do not occur. The question of whether Spec B is necessarily a framed curve (that is, free of multiple points) was raised by P. Perkins (see [18]) in a special case where the mad subalgebra B is dual (in the sense of Section 5 below) to O(X). He raised also a more subtle question: setting Y := Spec B, is it true that D(X) is isomorphic to D(Y )?…”

Rings of differential operators over rational affine curves

“…By [9, Proposition 3.11], it follows that 9-gr D(X) is a subring of C[x,y} and by [8,Lemma 2.3], x-gvD(X) is also a subring of C[x,y}. In the following lemma, we extend this to other gradings.…”

“…PERKINS studies rings that satisfy these conditions in [8]. He shows that in many cases, D(X) contains maximal commutative ad-nilpotent subalgebras not isomorphic to 0(X).…”

“…We will write 9-gr D(X) for giQ^D(X) and (9-deg for Vo,/3-Similarly, when f3 = 0 and a is positive the graded algebra determined by Va o is the same as rc-gr R determined by x-deg defined in [8].…”