On the Classification of Automorphic Lie Algebras

The problem of reduction of integrable equations can be formulated in a uniform way using the theory of invariants. This provides a powerful tool of analysis and it opens the road to new applications of Automorphic Lie Algebras, beyond the context of integrable systems. In this paper it is shown that Sl₂ℂ-based Automorphic Lie Algebras associated to the icosahedral group I, the octahedral group O, the tetrahedral group T, and the dihedral group D_n are isomorphic. The proof is based on techniques from classical invariant theory and makes use of Clebsch-Gordan decomposition and transvectants, Molien functions and the trace-form. This result provides a complete classification of Sl₂ℂ-based Automorphic Lie Algebras associated to finite groups when the group representations are chosen to be the same and it is a crucial step towards the complete classification of Automorphic Lie Algebras.