Zeitschrift für verallgemeinerte Lügentheorie und -anwendungen

Automorphic Lie Algebras in Higher Dimensions

Abstract

Him Cheung

The study offers the entire classification of automorphic Lie algebras based on sln(C), where sln(C) contains no trivial summands, the poles are in any of the exceptional G-orbits in C and the symmetry group G acts on sln(C) through inner automorphisms. The analysis of the algebras within the framework of traditional invariant theory is a crucial aspect of the categorization. This offers both a strong computational tool and raises new concerns from an algebraic standpoint (such as structure theory) that indicate to other uses for these algebras outside of the realm of integrable systems. The study demonstrates, in particular, that the class of automorphic Lie algebras connected to TOY groups (tetrahedral, octahedral and icosahedral groups) depend solely on the automorphic functions of the group, making them group independent Lie algebras. This may be proven by generalising the classical idea to the case of Lie algebras over a polynomial ring and creating a Chevalley normal form for these algebras.