Higher dimensional Automorphic Lie Algebras

TL;DR

This paper classifies Automorphic Lie Algebras based on sl_n(a9) with finite symmetry groups, revealing their structure via classical invariant theory and showing group independence for certain polyhedral groups.

Contribution

It provides a complete classification of these Lie algebras, introduces a Chevalley normal form over polynomial rings, and explores their algebraic and computational properties.

Findings

Automorphic Lie Algebras depend only on automorphic functions for T, O, Y groups.

Classification uses classical invariant theory as a computational tool.

Automorphic Lie Algebras are group independent for tetrahedral, octahedral, and icosahedral groups.

Abstract

The paper presents the complete classification of Automorphic Lie Algebras based on $\mathfrak{sl}_n (\mathbb{C})$, where the symmetry group $G$ is finite and the orbit is any of the exceptional $G$-orbits in $\overline{\mathbb{C}}$. A key feature of the classification is the study of the algebras in the context of classical invariant theory. This provides on one hand a powerful tool from the computational point of view, on the other it opens new questions from an algebraic perspective, which suggest further applications of these algebras, beyond the context of integrable systems. In particular, the research shows that Automorphic Lie Algebras associated to the $\mathbb{T}\mathbb{O}\mathbb{Y}$ groups (tetrahedral, octahedral and icosahedral groups) depend on the group through the automorphic functions only, thus they are group independent as Lie algebras. This can be established by defining a Chevalley normal form for these algebras, generalising this classical notion to the case of Lie algebras over a polynomial ring.