Wild Local Structures of Automorphic Lie Algebras

TL;DR

This paper investigates the complex structure of automorphic Lie algebras, revealing their wild representation type and linking their quotients to twisted polynomial current algebras, with connections to affine Kac-Moody algebras.

Contribution

It introduces a novel approach using evaluation maps to analyze automorphic Lie algebras and characterizes their local structures in terms of affine Kac-Moody algebras.

Findings

Automorphic Lie algebras are of wild representation type.

Quotients of these algebras are isomorphic to twisted truncated polynomial current algebras.

The local structure relates to affine Kac-Moody algebras.

Abstract

We study automorphic Lie algebras using a family of evaluation maps parametrised by the representations of the associative algebra of functions. This provides a descending chain of ideals for the automorphic Lie algebra which is used to prove that it is of wild representation type. We show that the associated quotients of the automorphic Lie algebra are isomorphic to twisted truncated polynomial current algebras. When a simple Lie algebra is used in the construction, this allows us to describe the local Lie structure of the automorphic Lie algebra in terms of affine Kac-Moody algebras.