A classification of automorphic Lie algebras on complex tori

TL;DR

This paper classifies automorphic Lie algebras on complex tori, identifying their structure and basis, and relates them to modular curves and Onsager's algebra, advancing understanding of symmetries in complex geometry.

Contribution

It provides a complete classification of automorphic Lie algebras on complex tori and describes their basis in a normal form, linking them to modular curves and known algebraic structures.

Findings

Automorphic Lie algebras form two disjoint families parametrized by the modular curve.

Four special cases are isomorphic to Onsager's algebra.

Explicit bases in normal form are computed for each case.

Abstract

We classify the automorphic Lie algebras of equivariant maps from a complex torus to $\mathfrak{sl}_2(\mathbb{C})$. For each case we compute a basis in a normal form. The automorphic Lie algebras correspond precisely to two disjoint families of Lie algebras parametrised by the modular curve of $\mathrm{PSL}_2(\mathbb{Z})$, apart from four cases, which are all isomorphic to Onsager's algebra.