Communications in Mathematical Physics

, Volume 299, Issue 3, pp 793–824 | Cite as

On the Classification of Automorphic Lie Algebras

Open Access


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.


Commutation Relation Dihedral Group Homogeneous Element Projective Representation Invariance Equation 
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The authors are grateful to A.V. Mikhailov for enlightening and fruitful discussions on various occasions. One of the authors, S. L., acknowledges financial support initially from EPSRC(EP/E044646/1) and then from NWO through the scheme VENI (016.073.026).

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© The Author(s) 2010

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of SciencesVrije UniversiteitAmsterdamThe Netherlands

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