Communications in Mathematical Physics

, Volume 267, Issue 3, pp 587–610 | Cite as

On \(\mathbb{Z}\)-Gradations of Twisted Loop Lie Algebras of Complex Simple Lie Algebras

  • Kh. S. NirovEmail author
  • A. V. Razumov


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.


Topological Vector Space Dimensional Manifold Loop Group Continuous Linear Mapping Grade Operator 
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© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Max-Planck-Institut für Gravitationsphysik – Albert-Einstein-InstitutGolm b. PotsdamGermany
  2. 2.Institute for High Energy PhysicsProtvino Moscow RegionRussia

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