Automorphic Lie Algebras and Modular Forms

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