Normal forms of elliptic automorphic Lie algebras and Landau-Lifshitz type of equations

2024/12/29 by Sara Lombardo, Lombardo, Sara, Casper Oelen +1
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Advanced Topics in Algebra #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.2412.20482

openalex publication_date 2024/12/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 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 \mathfraksl(2,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 \mathfraksl(2,ℂ)-current algebra and the two-dimensional abelian Lie algebra ℂ². 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.

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