A novel chain of Lie algebras and its coalgebra symmetry
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arXiv
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| Format: | Preprint |
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2025
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| author | Gubbiotti, Giorgio Latini, Danilo van Geemen, Bert |
| author_facet | Gubbiotti, Giorgio Latini, Danilo van Geemen, Bert |
| contents | We study a novel $n(n+1)/2$-dimensional non-semisimple Lie algebra $\mathfrak{g}_n$, a generalisation of both $\mathfrak{sl}_2(\mathbb{K})$ and the two-photon Lie algebra $\mathfrak{h}_6$. We investigate its properties, including its structure, representations, and its Casimir elements. In particular, we prove that there exists only one non-trivial Casimir polynomial of degree $n$ given by the determinant of an $n\times n$ symmetric matrix. We then associate this Lie algebra to a hierarchy of Hamiltonian systems with integrability properties depending on $n$, and describe their first integrals as sums of squares of linear combinations of the components of the angular momentum. In particular, we obtain that these systems are integrable for $n=2$, quasi-integrable for $n=3$, and of Poincaré-Lyapunov-Nekhoroshev type for $n\geq4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_01791 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A novel chain of Lie algebras and its coalgebra symmetry Gubbiotti, Giorgio Latini, Danilo van Geemen, Bert Mathematical Physics Representation Theory Exactly Solvable and Integrable Systems 17B80, 22E60 (Primary) 37J35, 70G65, 70H06 (Secondary) We study a novel $n(n+1)/2$-dimensional non-semisimple Lie algebra $\mathfrak{g}_n$, a generalisation of both $\mathfrak{sl}_2(\mathbb{K})$ and the two-photon Lie algebra $\mathfrak{h}_6$. We investigate its properties, including its structure, representations, and its Casimir elements. In particular, we prove that there exists only one non-trivial Casimir polynomial of degree $n$ given by the determinant of an $n\times n$ symmetric matrix. We then associate this Lie algebra to a hierarchy of Hamiltonian systems with integrability properties depending on $n$, and describe their first integrals as sums of squares of linear combinations of the components of the angular momentum. In particular, we obtain that these systems are integrable for $n=2$, quasi-integrable for $n=3$, and of Poincaré-Lyapunov-Nekhoroshev type for $n\geq4$. |
| title | A novel chain of Lie algebras and its coalgebra symmetry |
| topic | Mathematical Physics Representation Theory Exactly Solvable and Integrable Systems 17B80, 22E60 (Primary) 37J35, 70G65, 70H06 (Secondary) |
| url | https://arxiv.org/abs/2512.01791 |