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KAC-Moody Lie Algebras and the Classification of Nilpotent Lie Algebras of Maximal Rank

Published online by Cambridge University Press:  20 November 2018

L. J. Santharoubane
L. J. Santharoubane*
Affiliation:
University of Poitiers, Poitiers, Cedex, France
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Introduction. The natural problem of determining all the Lie algebras of finite dimension was broken in two parts by Levi's theorem:

1) the classification of semi-simple Lie algebras (achieved by Killing and Cartan around 1890)

2) the classification of solvable Lie algebras (reduced to the classification of nilpotent Lie algebras by Malcev in 1945 (see [10])).

The Killing form is identically equal to zero for a nilpotent Lie algebra but it is non-degenerate for a semi-simple Lie algebra. Therefore there was a huge gap between those two extreme cases. But this gap is only illusory because, as we will prove in this work, a large class of nilpotent Lie algebras is closely related to the Kac-Moody Lie algebras. These last algebras could be viewed as infinite dimensional version of the semisimple Lie algebras.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 34 , Issue 6 , 01 December 1982 , pp. 1215 - 1239
Copyright
Copyright © Canadian Mathematical Society 1982

References

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