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 $${\mathbb {C}}$$ 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 introduced by Holod, can each be realised as an elliptic $$\mathfrak {sl}(2,{\mathbb {C}})$$ sl ( 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 a complex Lie algebra, it is isomorphic to the direct sum of an $$\mathfrak {sl}(2,{\mathbb {C}})$$ sl ( 2 , C ) -current algebra and the two-dimensional abelian Lie algebra $${\mathbb {C}}^2$$ C 2 . Finally, we apply the automorphic Lie algebra framework to an n -component generalisation of the Landau–Lifshitz equation due to Golubchik and Sokolov in the case $$n=3$$ n = 3 .