A uniform construction of Chevalley normal forms for automorphic Lie algebras on the Riemann sphere
Vincent Knibbeler
TL;DR
The work develops a uniform, constructive framework for automorphic Lie algebras on the Riemann sphere by leveraging invariant theory of polyhedral groups, a universal intertwiner between scalar and vector invariants, and Chevalley-like normal forms for simple Lie algebras. It proves that even-degree invariants $R^{B\Gamma}_k$ are generated by a single element $P_k$ over the degree-zero invariants, with odd-degree invariants vanishing, enabling explicit generation of automorphic Lie algebras via $\omega^2$-twisted brackets. The authors classify SL$(2,\mathbb{C})$ embeddings into automorphism groups of $\mathfrak{g}$, construct an intertwiner $\Phi_{P,\bar{\rho}}$ yielding a graded-isomorphism to $\mathfrak{g}[X,Y,P^{-1}]^{B\Gamma}$, and provide Chevalley-normal-form bases with explicit structure constants for all exceptional types. They further illustrate the framework with rank-2 graphs, inner regular classifications for type $A_N$, and a complete data set of structure constants for exceptional Lie types, showing how isomorphisms across different symmetry groups arise from orbifold data and Coxeter numbers. The results offer a uniform, computable toolkit for constructing and comparing automorphic Lie algebras across simple Lie algebras and polyhedral symmetries, with potential implications for integrable systems and invariant-theoretic classifications.
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 $\mathbb{C}[X,Y,P^{-1}]^{G}_k$ of degree $k\in2\mathbb{Z}$ 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 $\mathfrak{g}[X,Y,P^{-1}]^{G}_0$ 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.