Realization of graded-simple algebras as loop algebras

Details Realization of graded-simple algebras as loop algebras Realization of graded-simple algebras as loop algebras

2005-11-30

arxiv.org/abs/math/0511723v2

Forum Math., 20(3):395-432,2008.

Mathematics

Rings and Algebras

31 pages. Corrected typos and added minor clarifications. Accepted in Forum Mathematicum

Scientific paper

Multiloop algebras determined by $n$ commuting algebra automorphisms of finite order are natural generalizations of the classical loop algebras that are used to realize affine Kac-Moody Lie algebras. In this paper, we obtain necessary and sufficient conditions for a $Z^n$-graded algebra to be realized as a multiloop algebra based on a finite dimensional simple algebra over an algebraically closed field of characteristic 0. We also obtain necessary and sufficient conditions for two such multiloop algebras to be graded-isomorphic, up to automorphism of the grading group. We prove these facts as consequences of corresponding results for a generalization of the multiloop construction. This more general setting allows us to work naturally and conveniently with arbitrary grading groups and arbitrary base fields.

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