Automorphic Lie Algebras with dihedral symmetry

TL;DR

This paper studies Automorphic Lie Algebras with dihedral symmetry, revealing a uniform structure across different pole configurations and orbit types, advancing understanding in integrable systems and symmetry reductions.

Contribution

It provides a unified description of Automorphic Lie Algebras with dihedral symmetry for various pole orbits and group actions, extending previous uniformity results.

Findings

Uniform description of Automorphic Lie Algebras with dihedral symmetry

Applicability to both exceptional and generic orbits

Enhanced understanding of symmetry in integrable systems

Abstract

The concept of Automorphic Lie Algebras arises in the context of reduction groups introduced in the early 1980s in the field of integrable systems. Automorphic Lie Algebras are obtained by imposing a discrete group symmetry on a current algebra of Krichever-Novikov type. Past work shows remarkable uniformity between algebras associated to different reduction groups. For example, if the base Lie algebra is $\mathfrak{sl}_2(\mathbb{C})$ and the poles of the Automorphic Lie Algebra are restricted to an exceptional orbit of the symmetry group, changing the reduction group does not affect the Lie algebra structure. In the present research we fix the reduction group to be the dihedral group and vary the orbit of poles as well as the group action on the base Lie algebra. We find a uniform description of Automorphic Lie Algebras with dihedral symmetry, valid for poles at exceptional and generic orbits.