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 $${\mathfrak{sl}_{2}(\mathbb{C})}$$-based Automorphic Lie Algebras associated to the icosahedral group $${{\mathbb I}}$$, the octahedral group $${{\mathbb O}}$$, the tetrahedral group $${{\mathbb T}}$$, and the dihedral group $${{\mathbb D}_n}$$ 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 $${\mathfrak{sl}_{2}(\mathbb{C})}$$-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.