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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2502.05137 |
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| _version_ | 1866909483195695104 |
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| author | Gubbiotti, Giorgio Oliveri, Francesco Sgroi, Emanuele Vergallo, Pierandrea |
| author_facet | Gubbiotti, Giorgio Oliveri, Francesco Sgroi, Emanuele Vergallo, Pierandrea |
| contents | We study from an algebraic and geometric viewpoint Hamiltonian operators which are sum of a non-degenerate first-order homogeneous operator and a Poisson tensor. In flat coordinates, also known as Darboux coordinates, these operators are uniquely determined by a triple composed by a Lie algebra, its most general non-degenerate quadratic Casimir and a 2-cocycle. We present some classes of operators associated to Lie algebras with non-degenerate quadratic Casimirs and we give a description of such operators in low dimensions. Finally, motivated by the example of the KdV equation we discuss the conditions of bi-Hamiltonianity of such operators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_05137 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lie algebras with compatible scalar products for non-homogeneous Hamiltonian operators Gubbiotti, Giorgio Oliveri, Francesco Sgroi, Emanuele Vergallo, Pierandrea Mathematical Physics Rings and Algebras We study from an algebraic and geometric viewpoint Hamiltonian operators which are sum of a non-degenerate first-order homogeneous operator and a Poisson tensor. In flat coordinates, also known as Darboux coordinates, these operators are uniquely determined by a triple composed by a Lie algebra, its most general non-degenerate quadratic Casimir and a 2-cocycle. We present some classes of operators associated to Lie algebras with non-degenerate quadratic Casimirs and we give a description of such operators in low dimensions. Finally, motivated by the example of the KdV equation we discuss the conditions of bi-Hamiltonianity of such operators. |
| title | Lie algebras with compatible scalar products for non-homogeneous Hamiltonian operators |
| topic | Mathematical Physics Rings and Algebras |
| url | https://arxiv.org/abs/2502.05137 |