Automorphic Lie algebras and corresponding integrable systems

TL;DR

This paper explores automorphic Lie algebras, generalizing Kac-Moody algebras, and demonstrates their application in formulating integrable systems, including hierarchies, Lax pairs, and symmetries.

Contribution

It classifies $sl(2,C)$ automorphic Lie algebras for all finite reduction groups and links these structures to integrable systems.

Findings

Classified automorphic Lie algebras for all finite reduction groups.

Connected automorphic Lie algebras to integrable hierarchies and Lax representations.

Established automorphic Lie algebras as a framework for integrable system symmetries.

Abstract

We study automorphic Lie algebras and their applications to integrable systems. Automorphic Lie algebras are a natural generalisation of celebrated Kac-Moody algebras to the case when the group of automorphisms is not cyclic. They are infinite dimensional and almost graded. We formulate the concept of a graded isomorphism and classify $sl(2,C)$ based automorphic Lie algebras corresponding to all finite reduction groups. We show that hierarchies of integrable systems, their Lax representations and master symmetries can be naturally formulated in terms of automorphic Lie algebras.