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

Abstract

Let ${\mathbb{M}}_n$ 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 ${\mathbb{M}}_1$ 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 ${\mathbb{M}}_1$. In this paper, we classify the algebras in ${\mathbb{M}}_2$, and further determine the relationship between ${\mathbb{M}}_2$ 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.