On the Classification of Automorphic Lie Algebras

Abstract

It is shown that the problem of reduction 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 these algebras, beyond the context of integrable systems. Moreover, it is proven that sl2-Automorphic Lie Algebras associated to the icosahedral group I, the octahedral group O, the tetrahedral group T, and the dihedral group Dn 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 sl2-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.

29 pages, 1 diagram, 9 tables, standard LaTeX2e, submitted for publication

References in corpus (3)

• Reduction Groups and Automorphic Lie Algebras
• Reductions of integrable equations. Dihedral group
• N-Wave Equations with Orthogonal Algebras: Z_2 and Z_2 \times Z_2 Reductions and Soliton Solutions

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• Higher dimensional Automorphic Lie Algebras
• Automorphic Lie Algebras with dihedral symmetry
• Rational Bundles and Recursion Operators for Integrable Equations on A.III-type Symmetric Spaces
• Darboux transformations with tetrahedral reduction group and related integrable systems
• Reductions of integrable equations on A.III-type symmetric spaces
• Automorphic Lie algebras and modular forms
• Darboux transformations, discrete integrable systems and related Yang-Baxter maps
• Hereditary Automorphic Lie Algebras
• Polyhedral Groups in $G_2(\mathbb{C})$
• A classification of automorphic Lie algebras on complex tori
• Computing equivariant matrices on homogeneous spaces for Geometric Deep Learning and Automorphic Lie Algebras
• A twisted integrable hierarchy with ${\mathbb{D}}_2$ symmetry