Rings of differential operators over rational affine curves

Letzter, Gail; Makar-Limanov, Leonid

doi:10.24033/bsmf.2143

Cited by 10 publications

(20 citation statements)

References 3 publications

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“…Our first result is as follows. This theorem is a sharper version of a result of [15], where it is shown that the normalization of Spec B is always isomorphic to A 1 ; that means that Spec B is obtained from A 1 by introducing cusps, but also, perhaps, identifying certain points of A 1 to form double points, or even higher order multiple points. The new part of Theorem 1.3 is thus the assertion that multiple points do not occur.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 85%

“…A detailed overview of the contents of the paper can be found in the introductory remarks to the individual sections that follow. Here we just mention that Sections 3 and 4 make no claim to originality, but are an exposition of some of the results of [15]. We give the exposition in some detail, because we rely heavily on these results, and the account of them given in [15] is not quite satisfying (specifically, part of Section 3 of that paper is missing, and the reference to [18] in the proof of its Corollary 4.6 seems difficult to justify directly).…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 95%

“…In this section we conclude our reworking of some parts of [15] which were not treated convincingly in that paper. The main results are Theorems 4.1 and 4.5.…”

Section: Rings With Several Mad Subalgebrasmentioning

confidence: 95%

“…The next two sections provide a self-contained exposition of some of the results of [15]. The present section gathers together some preliminary facts, culled from [12,15,17,19].…”

Section: Rings Of Differential Operatorsmentioning

confidence: 98%

“…The present section gathers together some preliminary facts, culled from [12,15,17,19]. The main points to note are Proposition 3.1, which ensures that the algebras studied later on all satisfy the condition (2.1); and the more technical-looking Proposition 3.5, which lies at the heart of the proofs in the following Section 4.…”

Section: Rings Of Differential Operatorsmentioning

confidence: 98%

See 4 more Smart Citations

Mad subalgebras of rings of differential operators on curves

Berest

Wilson²

2007

Advances in Mathematics

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Section: Introduction and Statement Of Resultsmentioning

confidence: 85%

Section: Introduction and Statement Of Resultsmentioning

confidence: 95%

“…In this section we conclude our reworking of some parts of [15] which were not treated convincingly in that paper. The main results are Theorems 4.1 and 4.5.…”

Section: Rings With Several Mad Subalgebrasmentioning

confidence: 95%

“…The next two sections provide a self-contained exposition of some of the results of [15]. The present section gathers together some preliminary facts, culled from [12,15,17,19].…”

Section: Rings Of Differential Operatorsmentioning

confidence: 98%

Section: Rings Of Differential Operatorsmentioning

confidence: 98%

See 3 more Smart Citations

Mad subalgebras of rings of differential operators on curves

Berest

Wilson²

2007

Advances in Mathematics

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Reflective prolate‐spheroidal operators and the adelic Grassmannian

Casper

Grünbaum

Yakimov

et al. 2023

Comm Pure Appl Math

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Beginning with the work of Landau, Pollak and Slepian in the 1960s on time‐band limiting, commuting pairs of integral and differential operators have played a key role in signal processing, random matrix theory, and integrable systems. Previously, such pairs were constructed by ad hoc methods, which essentially worked because a commuting operator of low order could be found by a direct calculation. We describe a general approach to these problems that proves that every point W of Wilson's infinite dimensional adelic Grassmannian gives rise to an integral operator , acting on for a contour , which reflects a differential operator with rational coefficients in the sense that on a dense subset of . By using analytic methods and methods from integrable systems, we show that the reflected differential operator can be constructed from the Fourier algebra of the associated bispectral function . The exact size of this algebra with respect to a bifiltration is in turn determined using algebro‐geometric methods. Intrinsic properties of four involutions of the adelic Grassmannian naturally lead us to consider the reflecting property above in place of plain commutativity. Furthermore, we prove that the time‐band limited operators of the generalized Laplace transforms with kernels given by the rank one bispectral functions always reflect a differential operator. A 90° rotation argument is used to prove that the time‐band limited operators of the generalized Fourier transforms with kernels admit a commuting differential operator. These methods produce vast collections of integral operators with prolate‐spheroidal properties, associated to the wave functions of all rational solutions of the KP hierarchy vanishing at infinity, introduced by Krichever in the late 1970s.

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Automorphisms and ideals of the Weyl algebra

Berest¹,

Wilson²

2000

Mathematische Annalen

126

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Let A 1 be the (first) Weyl algebra, and let G be its automorphism group. We study the natural action of G on the space of isomorphism classes of right ideals of A 1 (equivalently, of finitely generated rank 1 torsion-free right A 1 -modules). We show that this space breaks up into a countable number of orbits each of which is a finite dimensional algebraic variety. Our results are strikingly similar to those for the commutative algebra of polynomials in two variables; however, we do not know of any general principle that would allow us to predict this in advance. As a key step in the proof, we obtain a new description of the bispectral involution of [W1]. We also make some comments on the group G from the viewpoint of Shafarevich's theory of infinite dimensional algebraic groups.

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Rings of differential operators over rational affine curves

Cited by 10 publications

References 3 publications

Mad subalgebras of rings of differential operators on curves

Mad subalgebras of rings of differential operators on curves

Reflective prolate‐spheroidal operators and the adelic Grassmannian

Automorphisms and ideals of the Weyl algebra

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