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Sbornik: Mathematics, 2015, Volume 206, Issue 5, Pages 676–717
DOI: https://doi.org/10.1070/SM2015v206n05ABEH004475
(Mi sm8429)
 

This article is cited in 4 scientific papers (total in 4 papers)

Geometric properties of commutative subalgebras of partial differential operators

A. B. Zheglova, H. Kurkeb

a M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
b Humboldt University, Berlin
References:
Abstract: We investigate further algebro-geometric properties of commutative rings of partial differential operators, continuing our research started in previous articles. In particular, we start to explore the simplest and also certain known examples of quantum algebraically completely integrable systems from the point of view of a recent generalization of Sato's theory, developed by the first author. We give a complete characterization of the spectral data for a class of ‘trivial’ commutative algebras and strengthen geometric properties known earlier for a class of known examples. We also define a kind of restriction map from the moduli space of coherent sheaves with fixed Hilbert polynomial on a surface to an analogous moduli space on a divisor (both the surface and the divisor are part of the spectral data). We give several explicit examples of spectral data and corresponding algebras of commuting (completed) operators, producing as a by-product interesting examples of surfaces that are not isomorphic to spectral surfaces of any (maximal) commutative ring of partial differential operators of rank one. Finally, we prove that any commutative ring of partial differential operators whose normalization is isomorphic to the ring of polynomials $k[u,t]$ is a Darboux transformation of a ring of operators with constant coefficients.
Bibliography: 39 titles.
Keywords: commuting differential operators, quantum integrable systems, moduli space of coherent sheaves, Darboux transformation.
Funding agency Grant number
Russian Foundation for Basic Research 14-01-00178-а
13-01-00664
Ministry of Education and Science of the Russian Federation НШ-581.2014.1
This research was partially supported by the Russian Foundation for Basic Research (grant nos. 14-01-00178-a and 13-01-00664) and the Programme of the President of the Russian Federation for the Support of Leading Scientific Schools (grant no. НШ-581.2014.1).
Received: 06.10.2014 and 01.02.2015
Bibliographic databases:
Document Type: Article
UDC: 517.957+512.72+512.71
MSC: Primary 13N15, 37K20; Secondary 14H70
Language: English
Original paper language: Russian
Citation: A. B. Zheglov, H. Kurke, “Geometric properties of commutative subalgebras of partial differential operators”, Sb. Math., 206:5 (2015), 676–717
Citation in format AMSBIB
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\by A.~B.~Zheglov, H.~Kurke
\paper Geometric properties of commutative subalgebras of partial differential operators
\jour Sb. Math.
\yr 2015
\vol 206
\issue 5
\pages 676--717
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  • This publication is cited in the following 4 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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