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

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 $\mathbb{C}$. 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 $n=3$.