Reduction Groups and Automorphic Lie Algebras

TL;DR

This paper introduces automorphic Lie algebras, a new class of infinite-dimensional Lie algebras with applications to integrable equations, providing explicit bases and factorization properties.

Contribution

It constructs automorphic Lie algebras, details their quasigraded bases, explicit structure constants, and factorization into subalgebras, advancing the understanding of their algebraic structure.

Findings

Explicit bases for automorphic Lie algebras

Quasigraded structure with explicit structure constants

Factorization into two subalgebras similar to current algebras

Abstract

We study a new class of infinite dimensional Lie algebras, which has important applications to the theory of integrable equations. The construction of these algebras is very similar to the one for automorphic functions and this motivates the name automorphic Lie algebras. For automorphic Lie algebras we present bases in which they are quasigraded and all structure constants can be written out explicitly. These algebras have a useful factorisations on two subalgebras similar to the factorisation of the current algebra on the positive and negative parts.