A uniform construction of Chevalley normal forms for automorphic Lie algebras on the Riemann sphere

Abstract

For a finite subgroup $G$ of $SU(2)$ and one of its ground forms $P\in \mathbb{C}[X,Y]$, we show that the space of invariants $\mathbb{C}[X,Y,P^{-1}{]}_k^G$ of degree $k\in 2\mathbb{Z}$ is a cyclic module over the algebra of invariants of degree zero. We find a generator for this module, uniformly for all finite subgroups of $SU(2)$. Then we construct a uniform intertwiner sending the scalar invariants to vector-valued invariants. With these tools we construct all automorphic Lie algebras $\mathfrak{g}[X,Y,P^{-1}{]}_0^G$ defined by a homomorphism from the symmetry group $G$ into the automorphism group of a finite dimensional Lie algebra $\mathfrak{g}$, which factors through $SU(2)$. When the Lie algebra $\mathfrak{g}$ is simple, we present a set of generators for the automorphic Lie algebra which is analogous to the Chevalley basis for $\mathfrak{g}$. Previous observations of isomorphisms between automorphic Lie algebras with distinct symmetry groups $G$ are explained in terms of the Coxeter number of $\mathfrak{g}$ and the orders appearing in $G$. Finally, we compute the structure constants for automorphic Lie algebras of all exceptional Lie types.