Automorphic Lie Algebras and Modular Forms

We introduce and study certain hyperbolic versions of automorphic Lie algebras related to the modular group. Let Γ be a finite index subgroup of SL (2, Z) with an action on a complex simple Lie algebra g, which can be extended to SL(2, C). We show that the Lie algebra of the corresponding g-valued modular forms is isomorphic to the extension of g over the usual modular forms. This establishes a modular analogue of a well-known result by Kac on twisted loop algebras. The case of principal congruence subgroups Γ (N), \, N≤ 6, is considered in more detail in relation to the classical results of Klein and Fricke and the celebrated Markov Diophantine equation. We finish with a brief discussion of the extensions and representations of these Lie algebras.