Zhukovsky-Volterra top and quantisation ideals
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910460108865536 |
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| author | Mikhailov, A. Skrypnyk, T. |
| author_facet | Mikhailov, A. Skrypnyk, T. |
| contents | In this letter, we revisit the quantisation problem for a fundamental model of classical mechanics - the Zhukovsky-Volterra top. We have discovered a four-parametric pencil of compatible Poisson brackets, comprising two quadratic and two linear Poisson brackets. Using the quantisation ideal method, we have identified two distinct quantisations of the Zhukovsky-Volterra top. The first type corresponds to the universal enveloping algebras of $so(3)$, leading to Lie-Poisson brackets in the classical limit. The second type can be regarded as a quantisation of the four-parametric inhomogeneous quadratic Poisson pencil. We discuss the relationships between the quantisations obtained in our paper, Sklyanin's quantisation of the Euler top, and Levin-Olshanetsky-Zotov's quantisation of the Zhukovsky-Volterra top. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_16532 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Zhukovsky-Volterra top and quantisation ideals Mikhailov, A. Skrypnyk, T. Exactly Solvable and Integrable Systems Mathematical Physics Quantum Physics In this letter, we revisit the quantisation problem for a fundamental model of classical mechanics - the Zhukovsky-Volterra top. We have discovered a four-parametric pencil of compatible Poisson brackets, comprising two quadratic and two linear Poisson brackets. Using the quantisation ideal method, we have identified two distinct quantisations of the Zhukovsky-Volterra top. The first type corresponds to the universal enveloping algebras of $so(3)$, leading to Lie-Poisson brackets in the classical limit. The second type can be regarded as a quantisation of the four-parametric inhomogeneous quadratic Poisson pencil. We discuss the relationships between the quantisations obtained in our paper, Sklyanin's quantisation of the Euler top, and Levin-Olshanetsky-Zotov's quantisation of the Zhukovsky-Volterra top. |
| title | Zhukovsky-Volterra top and quantisation ideals |
| topic | Exactly Solvable and Integrable Systems Mathematical Physics Quantum Physics |
| url | https://arxiv.org/abs/2405.16532 |