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

2025/03/22 by Knibbeler, Vincent
#13A50 (Secondary) #17B65 (Primary) 17B05 #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2503.17801

Abstract

For a finite subgroup G of SU(2) and one of its ground forms P∈ℂ[X,Y], we show that the space of invariants ℂ[X,Y,P^-1]^G_k of degree k∈2ℤ 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 \mathfrakg[X,Y,P^-1]^G₀ 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.

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