Poisson Algebras and 3D Superintegrable Hamiltonian Systems
Using a Poisson bracket representation, in 3D, of the Lie algebra sl(2), we first use highest weight representations to embed this into larger Lie algebras. These are then interpreted as symmetry and conformal symmetry algebras of the "kinetic energy", related to the quadratic Casimir func...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2018 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2018
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/209442 |
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| Cite this: | Poisson Algebras and 3D Superintegrable Hamiltonian Systems / A.P. Fordy, Q. Huang // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 16 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
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Fordy, A.P. Huang, Q. 2025-11-21T18:55:21Z 2018 Poisson Algebras and 3D Superintegrable Hamiltonian Systems / A.P. Fordy, Q. Huang // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 16 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 17B63; 37J15; 37J35; 70G45; 70G65; 70H06 arXiv: 1708.07024 https://nasplib.isofts.kiev.ua/handle/123456789/209442 https://doi.org/10.3842/SIGMA.2018.022 Using a Poisson bracket representation, in 3D, of the Lie algebra sl(2), we first use highest weight representations to embed this into larger Lie algebras. These are then interpreted as symmetry and conformal symmetry algebras of the "kinetic energy", related to the quadratic Casimir function. We then consider the potentials which can be added, whilst remaining integrable, leading to families of separable systems, depending upon arbitrary functions of a single variable. Adding further integrals, in the superintegrable case, restricts these functions to specific forms, depending upon a finite number of arbitrary parameters. The Poisson algebras of these superintegrable systems are studied. The automorphisms of the symmetry algebra of the kinetic energy are extended to the full Poisson algebra, enabling us to build the full set of Poisson relations. This work was carried out while QH was visiting Leeds for one year, funded by the China Scholarship Council. QH would like to thank the School of Mathematics, University of Leeds, for its hospitality. Significant improvements were made after the comments of both referees and the editor. We thank them for their input. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Poisson Algebras and 3D Superintegrable Hamiltonian Systems Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| title |
Poisson Algebras and 3D Superintegrable Hamiltonian Systems |
| spellingShingle |
Poisson Algebras and 3D Superintegrable Hamiltonian Systems Fordy, A.P. Huang, Q. |
| title_short |
Poisson Algebras and 3D Superintegrable Hamiltonian Systems |
| title_full |
Poisson Algebras and 3D Superintegrable Hamiltonian Systems |
| title_fullStr |
Poisson Algebras and 3D Superintegrable Hamiltonian Systems |
| title_full_unstemmed |
Poisson Algebras and 3D Superintegrable Hamiltonian Systems |
| title_sort |
poisson algebras and 3d superintegrable hamiltonian systems |
| author |
Fordy, A.P. Huang, Q. |
| author_facet |
Fordy, A.P. Huang, Q. |
| publishDate |
2018 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
Using a Poisson bracket representation, in 3D, of the Lie algebra sl(2), we first use highest weight representations to embed this into larger Lie algebras. These are then interpreted as symmetry and conformal symmetry algebras of the "kinetic energy", related to the quadratic Casimir function. We then consider the potentials which can be added, whilst remaining integrable, leading to families of separable systems, depending upon arbitrary functions of a single variable. Adding further integrals, in the superintegrable case, restricts these functions to specific forms, depending upon a finite number of arbitrary parameters. The Poisson algebras of these superintegrable systems are studied. The automorphisms of the symmetry algebra of the kinetic energy are extended to the full Poisson algebra, enabling us to build the full set of Poisson relations.
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| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/209442 |
| citation_txt |
Poisson Algebras and 3D Superintegrable Hamiltonian Systems / A.P. Fordy, Q. Huang // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 16 назв. — англ. |
| work_keys_str_mv |
AT fordyap poissonalgebrasand3dsuperintegrablehamiltoniansystems AT huangq poissonalgebrasand3dsuperintegrablehamiltoniansystems |
| first_indexed |
2025-12-07T21:04:44Z |
| last_indexed |
2025-12-07T21:04:44Z |
| _version_ |
1850886155968446464 |