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

### Arturo Pianzola

University of Alberta, Edmonton, Canada### Bruce Allison

University of Alberta, Edmonton, Canada### Stephen Berman

Saskatoon, Canada

## 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.

## Cite this article

Arturo Pianzola, Bruce Allison, Stephen Berman, Multiloop algebras, iterated loop algebras and extended affine Lie algebras of nullity 2. J. Eur. Math. Soc. 16 (2014), no. 2, pp. 327–385

DOI 10.4171/JEMS/435