Abstract
AbstractWe introduce and study certain hyperbolic versions of automorphic Lie algebras related to the modular group. Let $\Gamma $ be a finite index subgroup of $\textrm {SL}(2,\mathbb Z)$ with an action on a complex simple Lie algebra $\mathfrak g$, which can be extended to $\textrm {SL}(2,{\mathbb {C}})$. We show that the Lie algebra of the corresponding $\mathfrak {g}$-valued modular forms is isomorphic to the extension of $\mathfrak {g}$ 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 $\Gamma (N), \, N\leq 6$, 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.
Highlights
Let g be a simple finite-dimensional complex Lie algebra and g ⊗C C[z, z−1] be its loop algebra consisting of Laurent polynomials of z with values in g
We have found a complete description of the automorphic Lie algebras on the upper half plane for representations ρ : Γ → Aut(g) factoring through SL(2, C)
A interesting class is given by Weil representations of Γ(1) representations from the theory of Jacobi modular forms, which can be naturally interpreted as the corresponding vector-valued modular forms
Summary
Let g be a simple finite-dimensional complex Lie algebra and g ⊗C C[z, z−1] be its loop algebra consisting of Laurent polynomials of z with values in g. Let, as before, g be a complex finite-dimensional simple Lie algebra and G be a corresponding Lie group, which in this case can be chosen as the connected component Aut(g)0 of the automorphism group of g. We consider the simplest example of our automorphic Lie algebras when Γ = SL(2, Z) is the modular group itself acting by conjugation on g = sl(2, C). We discuss the theory of vector-valued modular forms (VVMF) for all finite index subgroups of modular groups and the particular class of representations restricted from irreducible representations of SL(2, C). We finish with a brief discussion of the extensions and representations of our algebras
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