On $mathbb{Z}$ -Gradations of Twisted Loop Lie Algebras of Complex Simple Lie Algebras

摘要

We define the twisted loop Lie algebra of a finite dimensional Lie algebra $mathfrak{g}$ as the Fréchet 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 Fréchet Lie algebras. We introduce the notion of an integrable $mathbb{Z}$ -gradation of a Fréchet 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.