Multiloop algebras, iterated loop algebras and extended affine Lie algebras of nullity 2

TL;DR

This paper classifies multiloop algebras of nullity 2, relating them to affine and extended affine Lie algebras, extending classical results from the nullity 1 case.

Contribution

It provides a classification of all multiloop algebras of nullity 2 and explores their relationship with affine and extended affine Lie algebras.

Findings

Classified all multiloop algebras of nullity 2.

Established connections between M_2, loop algebras of affine Lie algebras, and extended affine Lie algebras.

Extended the classification framework from nullity 1 to nullity 2.

Abstract

Let M_n be the class of all multiloop algebras of finite dimensional simple Lie algebras relative to n-tuples of commuting finite order automorphisms. It is a classical result that M_1 is the class of all derived algebras modulo their centres of affine Kac-Moody Lie algebras. This combined with the Peterson-Kac conjugacy theorem for affine algebras results in a classification of the algebras in M_1. In this paper, we classify the algebras in M_2, and further determine the relationship between M_2 and two other classes of Lie algebras: the class of all loop algebras of affine Lie algebras and the class of all extended affine Lie algebras of nullity 2.