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Realization of graded-simple algebras as loop algebras

Realization of graded-simple algebras as loop algebras 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. 2000 Mathematics Subject Classification: 16W50, 17B70; 17B65, 17B67. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

Realization of graded-simple algebras as loop algebras

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Publisher
de Gruyter
Copyright
© Walter de Gruyter
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/FORUM.2008.020
Publisher site
See Article on Publisher Site

Abstract

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. 2000 Mathematics Subject Classification: 16W50, 17B70; 17B65, 17B67.

Journal

Forum Mathematicumde Gruyter

Published: May 1, 2008

References