(3) Polynomial Integrable System: Different Faces of the 3-Body/₂ Elliptic Calogero Model
It is shown that the (3) polynomial integrable system, introduced by Sokolov-Turbiner in [J. Phys. A 48 (2015), 155201, 15 pages, arXiv:1409.7439], is equivalent to the (3) quantum Euler-Arnold top in a constant magnetic field. Their Hamiltonian and third-order integral can be rewritten in terms of...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2024 |
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Інститут математики НАН України
2024
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| Zitieren: | (3) Polynomial Integrable System: Different Faces of the 3-Body/₂ Elliptic Calogero Model. Alexander V. Turbiner, Juan Carlos Lopez Vieyra and Miguel A. Guadarrama-Ayala. SIGMA 20 (2024), 012, 23 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862596236270370816 |
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| author | Turbiner, Alexander V. Lopez Vieyra, Juan Carlos Guadarrama-Ayala, Miguel A. |
| author_facet | Turbiner, Alexander V. Lopez Vieyra, Juan Carlos Guadarrama-Ayala, Miguel A. |
| citation_txt | (3) Polynomial Integrable System: Different Faces of the 3-Body/₂ Elliptic Calogero Model. Alexander V. Turbiner, Juan Carlos Lopez Vieyra and Miguel A. Guadarrama-Ayala. SIGMA 20 (2024), 012, 23 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | It is shown that the (3) polynomial integrable system, introduced by Sokolov-Turbiner in [J. Phys. A 48 (2015), 155201, 15 pages, arXiv:1409.7439], is equivalent to the (3) quantum Euler-Arnold top in a constant magnetic field. Their Hamiltonian and third-order integral can be rewritten in terms of the (3) algebra generators. In turn, all these (3) generators can be represented by the non-linear elements of the universal enveloping algebra of the 5-dimensional Heisenberg algebra ₅(^₁,₂, ^₁,₂, ); thus, the Hamiltonian and integral are two elements of the universal enveloping algebra ₅. In this paper, four different representations of the ₅ Heisenberg algebra are used: (I) by differential operators in two real (complex) variables, (II) by finite-difference operators on uniform or exponential lattices. We discovered the existence of two 2-parametric bilinear and trilinear elements (denoted and , respectively) of the universal enveloping algebra ((3)) such that their Lie bracket (commutator) can be written as a linear superposition of nine so-called artifacts - the special bilinear elements of ((3)), which vanish once the representation of the (3)-algebra generators is written in terms of the ₅(^₁,₂, ^₁,₂, )-algebra generators. In this representation, all nine artifacts vanish, two of the above-mentioned elements of U((3)) (called the Hamiltonian H and the integral I) commute(!); in particular, they become the Hamiltonian and the integral of the 3-body elliptic Calogero model, if (^, ^) are written in the standard coordinate-momentum representation. If (^, ^) are represented by finite-difference/discrete operators on uniform or exponential lattices, the Hamiltonian and the integral of the 3-body elliptic Calogero model become the isospectral, finite-difference operators on uniform-uniform or exponential-exponential lattices (or mixed) with polynomial coefficients. If (^, ^) are written in complex (, ¯) variables, the Hamiltonian corresponds to a complexification of the 3-body elliptic Calogero model on ℂ².
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| first_indexed | 2026-03-13T23:28:54Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-212111 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-13T23:28:54Z |
| publishDate | 2024 |
| publisher | Інститут математики НАН України |
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| spelling | Turbiner, Alexander V. Lopez Vieyra, Juan Carlos Guadarrama-Ayala, Miguel A. 2026-01-28T13:56:39Z 2024 (3) Polynomial Integrable System: Different Faces of the 3-Body/₂ Elliptic Calogero Model. Alexander V. Turbiner, Juan Carlos Lopez Vieyra and Miguel A. Guadarrama-Ayala. SIGMA 20 (2024), 012, 23 pages 1815-0659 2020 Mathematics Subject Classification: 81R12; 81S05; 17J35; 81U15 arXiv:2305.00529 https://nasplib.isofts.kiev.ua/handle/123456789/212111 https://doi.org/10.3842/SIGMA.2024.012 It is shown that the (3) polynomial integrable system, introduced by Sokolov-Turbiner in [J. Phys. A 48 (2015), 155201, 15 pages, arXiv:1409.7439], is equivalent to the (3) quantum Euler-Arnold top in a constant magnetic field. Their Hamiltonian and third-order integral can be rewritten in terms of the (3) algebra generators. In turn, all these (3) generators can be represented by the non-linear elements of the universal enveloping algebra of the 5-dimensional Heisenberg algebra ₅(^₁,₂, ^₁,₂, ); thus, the Hamiltonian and integral are two elements of the universal enveloping algebra ₅. In this paper, four different representations of the ₅ Heisenberg algebra are used: (I) by differential operators in two real (complex) variables, (II) by finite-difference operators on uniform or exponential lattices. We discovered the existence of two 2-parametric bilinear and trilinear elements (denoted and , respectively) of the universal enveloping algebra ((3)) such that their Lie bracket (commutator) can be written as a linear superposition of nine so-called artifacts - the special bilinear elements of ((3)), which vanish once the representation of the (3)-algebra generators is written in terms of the ₅(^₁,₂, ^₁,₂, )-algebra generators. In this representation, all nine artifacts vanish, two of the above-mentioned elements of U((3)) (called the Hamiltonian H and the integral I) commute(!); in particular, they become the Hamiltonian and the integral of the 3-body elliptic Calogero model, if (^, ^) are written in the standard coordinate-momentum representation. If (^, ^) are represented by finite-difference/discrete operators on uniform or exponential lattices, the Hamiltonian and the integral of the 3-body elliptic Calogero model become the isospectral, finite-difference operators on uniform-uniform or exponential-exponential lattices (or mixed) with polynomial coefficients. If (^, ^) are written in complex (, ¯) variables, the Hamiltonian corresponds to a complexification of the 3-body elliptic Calogero model on ℂ². A.V.T. is thankful to Willard Miller Jr. (1937–2023) and Peter Olver (University of Minnesota, USA) for helpful discussions at different stages of the project and for their general encouragement to proceed and complete this work. Due to enormous computational complexity, this research has been running for many years. It was supported in part by the PAPIIT grants IN109512 and IN108815 (Mexico) at the initial stage of the study and by the PAPIIT grant IN113022 (Mexico) at its final stage. M.A.G.A. thanks the CONACyT grant for master's degree studies (Mexico) in 2016–2018, when the key calculations of the commutator (6.3) were partially carried out. All symbolic calculations were carried out by using the MAPLE-18 on a regular DELL desktop computer with a CPU processor Intel(R) Core (TM) i7-3770 @ 3.40GHz with 6Gb RAM, although some pieces of calculations were done on a regular PC laptop. A.V.T. thanks the PASPA-UNAM grant (Mexico) for its support during his sabbatical stay in 2021–2022 at the University of Miami, where this work was mostly completed. We thank all anonymous referees for careful reading of the text, many inspiring comments, remarks, and proposals, which improved significantly the presentation. This work is dedicated to the 70th birthday of Peter Olver, to whom we always admired as an exemplary mathematician and scientist. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications (3) Polynomial Integrable System: Different Faces of the 3-Body/₂ Elliptic Calogero Model Article published earlier |
| spellingShingle | (3) Polynomial Integrable System: Different Faces of the 3-Body/₂ Elliptic Calogero Model Turbiner, Alexander V. Lopez Vieyra, Juan Carlos Guadarrama-Ayala, Miguel A. |
| title | (3) Polynomial Integrable System: Different Faces of the 3-Body/₂ Elliptic Calogero Model |
| title_full | (3) Polynomial Integrable System: Different Faces of the 3-Body/₂ Elliptic Calogero Model |
| title_fullStr | (3) Polynomial Integrable System: Different Faces of the 3-Body/₂ Elliptic Calogero Model |
| title_full_unstemmed | (3) Polynomial Integrable System: Different Faces of the 3-Body/₂ Elliptic Calogero Model |
| title_short | (3) Polynomial Integrable System: Different Faces of the 3-Body/₂ Elliptic Calogero Model |
| title_sort | (3) polynomial integrable system: different faces of the 3-body/₂ elliptic calogero model |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212111 |
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