Communications in Mathematical Physics

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

On the Classification of Automorphic Lie Algebras

  • Sara LombardoEmail author
  • Jan A. Sanders
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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



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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This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution,and reproduction in any medium, provided the original author(s) and source are credited.


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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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