On Z-gradations of twisted loop Lie algebras of complex simple Lie algebras

TL;DR

This paper characterizes all inequivalent integrable Z-gradations with finite-dimensional grading subspaces in twisted loop Lie algebras of complex simple Lie algebras, expanding understanding of their structure.

Contribution

It introduces the concept of integrable Z-gradations in Fréchet Lie algebras and classifies all such gradations with finite-dimensional subspaces for twisted loop Lie algebras.

Findings

All inequivalent integrable Z-gradations with finite-dimensional grading subspaces are classified.

The structure of twisted loop Lie algebras of complex simple Lie algebras is elucidated.

The notion of integrable Z-gradation in Fréchet Lie algebras is established.

Abstract

We define the twisted loop Lie algebra of a finite dimensional Lie algebra $\mathfrak g$ as the Fr\'echet space of all twisted periodic smooth mappings from $\mathbb R$ to $\mathfrak g$. Here the Lie algebra operation is continuous. We call such Lie algebras Fr\'echet Lie algebras. We introduce the notion of an integrable $\mathbb Z$-gradation of a Fr\'echet Lie algebra, and find all inequivalent integrable $\mathbb Z$-gradations with finite dimensional grading subspaces of twisted loop Lie algebras of complex simple Lie algebras.