Hereditary Automorphic Lie Algebras

TL;DR

This paper classifies hereditary Automorphic Lie Algebras using 2-cocycles on root systems, linking their structure to equivariant vectors and automorphic functions.

Contribution

It introduces a novel classification approach for hereditary Automorphic Lie Algebras via 2-cocycles and provides methods to compute their structure and automorphic functions.

Findings

Hereditary Automorphic Lie Algebras are characterized by 2-cocycles with two values.

The Killing form and abelianisation can be derived from cocycle values.

A reduction method simplifies the computation of equivariant vectors.

Abstract

We show that Automorphic Lie Algebras which contain a Cartan subalgebra with a constant spectrum, called hereditary, are completely described by 2-cocycles on a classical root system taking only two different values. This observation suggests a novel approach to their classification. By determining the values of the cocycles on opposite roots, we obtain the Killing form and the abelianisation of the Automorphic Lie Algebra. The results are obtained by studying equivariant vectors on the projective line. As a byproduct, we describe a method to reduce the computation of the infinite dimensional space of said equivariant vectors to a finite dimensional linear computation and the determination of the ring of automorphic functions on the projective line.