Automorphic Lie algebras on complex tori

An automorphic Lie algebra is a Lie algebra of certain invariants, initially arising in the theory of integrable systems, or more specifically, in the context of algebraic reduction of Lax pairs. They are defined as follows. Let a finite group Γ act on a compact Riemann surface and on a complex finite dimensional Lie algebra g, both by automorphisms. Consider the space of meromorphic maps from the Riemann surface to the Lie algebra with poles restricted to an orbit of Γ. The subspace of Γ-equivariant maps is an automorphic Lie algebra. It is an infinite dimensional Lie algebra over the complex numbers and it can be seen as a generalisation of the Onsager algebra. We lay out a classification programme for automorphic Lie algebras on complex tori and present a classification in the case of g = sl₂. One of the main contributions is the construction of normal forms for automorphic Lie algebras on complex tori with g = sl₂, in which case Γ is either cyclic, dihedral or the alternating group on four symbols.