Commuting Ordinary Differential Operators and the Dixmier Test
The Burchnall-Chaundy problem is classical in differential algebra, seeking to describe all commutative subalgebras of a ring of ordinary differential operators whose coefficients are functions in a given class. It received less attention when posed in the (first) Weyl algebra, namely for polynomial...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2019 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2019
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/210287 |
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| Cite this: | Commuting Ordinary Differential Operators and the Dixmier Test / E. Previato, S.L. Rueda, M.-A. Zurro // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 46 назв. — англ. |
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Previato, E. Rueda, S.L. Zurro, M.-A. 2025-12-05T09:21:07Z 2019 Commuting Ordinary Differential Operators and the Dixmier Test / E. Previato, S.L. Rueda, M.-A. Zurro // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 46 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 13P15; 14H70 arXiv: 1902.01361 https://nasplib.isofts.kiev.ua/handle/123456789/210287 https://doi.org/10.3842/SIGMA.2019.101 The Burchnall-Chaundy problem is classical in differential algebra, seeking to describe all commutative subalgebras of a ring of ordinary differential operators whose coefficients are functions in a given class. It received less attention when posed in the (first) Weyl algebra, namely for polynomial coefficients, while the classification of commutative subalgebras of the Weyl algebra is in itself an important open problem. Centralizers are maximal-commutative subalgebras, and we review the properties of a basis of the centralizer of an operator L in normal form, following the approach of K.R. Goodearl, with the ultimate goal of obtaining such bases by computational routines. Our first step is to establish the Dixmier test, based on a lemma by J. Dixmier and the choice of a suitable filtration, to give necessary conditions for an operator M to be in the centralizer of L. Whenever the centralizer equals the algebra generated by L and M, we call L, M a Burchnall-Chaundy (BC) pair. A construction of BC pairs is presented for operators of order 4 in the first Weyl algebra. Moreover, for true rank r pairs, by means of differential subresultants, we effectively compute the fiber of the rank r spectral sheaf over their spectral curve. The authors would like to thank the organizers of the conference AMDS2018, which took place in Madrid, for giving them the opportunity to collaborate on these topics of common interest for a long time, and finally write this paper together. The authors would like to thank the anonymous referees who have helped to improve the final version of this work. S.L. Rueda is partially supported by the Research Group "Modelos matemáticos no lineales". M.A. Zurro is partially supported by Grupo UCM 910444. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Commuting Ordinary Differential Operators and the Dixmier Test Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| title |
Commuting Ordinary Differential Operators and the Dixmier Test |
| spellingShingle |
Commuting Ordinary Differential Operators and the Dixmier Test Previato, E. Rueda, S.L. Zurro, M.-A. |
| title_short |
Commuting Ordinary Differential Operators and the Dixmier Test |
| title_full |
Commuting Ordinary Differential Operators and the Dixmier Test |
| title_fullStr |
Commuting Ordinary Differential Operators and the Dixmier Test |
| title_full_unstemmed |
Commuting Ordinary Differential Operators and the Dixmier Test |
| title_sort |
commuting ordinary differential operators and the dixmier test |
| author |
Previato, E. Rueda, S.L. Zurro, M.-A. |
| author_facet |
Previato, E. Rueda, S.L. Zurro, M.-A. |
| publishDate |
2019 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
The Burchnall-Chaundy problem is classical in differential algebra, seeking to describe all commutative subalgebras of a ring of ordinary differential operators whose coefficients are functions in a given class. It received less attention when posed in the (first) Weyl algebra, namely for polynomial coefficients, while the classification of commutative subalgebras of the Weyl algebra is in itself an important open problem. Centralizers are maximal-commutative subalgebras, and we review the properties of a basis of the centralizer of an operator L in normal form, following the approach of K.R. Goodearl, with the ultimate goal of obtaining such bases by computational routines. Our first step is to establish the Dixmier test, based on a lemma by J. Dixmier and the choice of a suitable filtration, to give necessary conditions for an operator M to be in the centralizer of L. Whenever the centralizer equals the algebra generated by L and M, we call L, M a Burchnall-Chaundy (BC) pair. A construction of BC pairs is presented for operators of order 4 in the first Weyl algebra. Moreover, for true rank r pairs, by means of differential subresultants, we effectively compute the fiber of the rank r spectral sheaf over their spectral curve.
|
| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/210287 |
| citation_txt |
Commuting Ordinary Differential Operators and the Dixmier Test / E. Previato, S.L. Rueda, M.-A. Zurro // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 46 назв. — англ. |
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AT previatoe commutingordinarydifferentialoperatorsandthedixmiertest AT ruedasl commutingordinarydifferentialoperatorsandthedixmiertest AT zurroma commutingordinarydifferentialoperatorsandthedixmiertest |
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2025-12-07T21:25:02Z |
| last_indexed |
2025-12-07T21:25:02Z |
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1850886274556100608 |