Normal forms of elliptic automorphic Lie algebras and Landau-Lifshitz type of equations
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909958083182592 |
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| author | Lombardo, Sara Oelen, Casper |
| author_facet | Lombardo, Sara Oelen, Casper |
| contents | We present normal forms of elliptic automorphic Lie algebras with dihedral symmetry of order 4, which arise naturally in the context of Landau-Lifshitz type of equations. These normal forms provide a transparent description and allow a classification of such Lie algebras over $\mathbb{C}$. Using this perspective, we show that a Lie algebra introduced by Uglov, as well as the hidden symmetry algebra of the Landau-Lifshitz equation by Holod, are both isomorphic to an elliptic $\mathfrak{sl}(2,\mathbb{C})$-current algebra. Furthermore, we realise the Wahlquist-Estabrook algebra of the Landau-Lifshitz equation in terms of elliptic automorphic Lie algebras. This construction reveals that, as complex Lie algebras, it is isomorphic to the direct sum of an $\mathfrak{sl}(2,\mathbb{C})$-current algebra and the two-dimensional abelian Lie algebra $\mathbb{C}^2$. Finally, we explicitly implement the automorphic Lie algebra framework in the context of an $n$-component generalisation of the Landau-Lifshitz equation by Golubchik and Sokolov in the case of $n=3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_20482 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Normal forms of elliptic automorphic Lie algebras and Landau-Lifshitz type of equations Lombardo, Sara Oelen, Casper Rings and Algebras Mathematical Physics We present normal forms of elliptic automorphic Lie algebras with dihedral symmetry of order 4, which arise naturally in the context of Landau-Lifshitz type of equations. These normal forms provide a transparent description and allow a classification of such Lie algebras over $\mathbb{C}$. Using this perspective, we show that a Lie algebra introduced by Uglov, as well as the hidden symmetry algebra of the Landau-Lifshitz equation by Holod, are both isomorphic to an elliptic $\mathfrak{sl}(2,\mathbb{C})$-current algebra. Furthermore, we realise the Wahlquist-Estabrook algebra of the Landau-Lifshitz equation in terms of elliptic automorphic Lie algebras. This construction reveals that, as complex Lie algebras, it is isomorphic to the direct sum of an $\mathfrak{sl}(2,\mathbb{C})$-current algebra and the two-dimensional abelian Lie algebra $\mathbb{C}^2$. Finally, we explicitly implement the automorphic Lie algebra framework in the context of an $n$-component generalisation of the Landau-Lifshitz equation by Golubchik and Sokolov in the case of $n=3$. |
| title | Normal forms of elliptic automorphic Lie algebras and Landau-Lifshitz type of equations |
| topic | Rings and Algebras Mathematical Physics |
| url | https://arxiv.org/abs/2412.20482 |