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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

1. Amiguet, D., Extensions inessentielles d'algèbres de Lie à noyau nilpotent, Thèse (1971), Ecole Polytechnique Fédérale de Lausanne.Google Scholar
2. Bourbaki, , Groupes etalgèbres de Lie, Chapter 1 (Hermann, Paris, 1968).Google Scholar
3. Bourbaki, , Groupes et algèbres de Lie, Chapters 2, 3 (Hermann, Paris, 1968).Google Scholar
4. Dixmier, , Sur les représentations unitaires des groupes de Lie nilpotentes III, Can. J. Math. 10 (1958), 321–348.Google Scholar
5. Favre, F., Système de poids sur une algèbre de Lie nilpotente, Manuscripta Math. 9 (1973), 53–90.Google Scholar
6. Gauger, M. A., On the classification of metabelian Lie algebras, Trans. Amer. Math. Soc. 179 (1973), 293–329.Google Scholar
7. Humpreys, J. E., Introduction to Lie algebras and representation theory (Springer- Verlag).Google Scholar
8. Kac, V. G., Simple irreducible graded Lie algebras of finite growth, Math. U.S.S.R. Izvestija 2 (1968), 1271–1311.Google Scholar
9. Lepowsky, J. and Milne, S., Lie algebraic approaches to classical partition identities, Advances in Math. 29 (1978), 15–59.Google Scholar
10. Malcev, A. I., Solvable Lie algebras, Amer. Math. Soc. Transi. (1) 9 (1962), 228–262.Google Scholar
11. Moody, R. V., A new class of Lie algebras, Journal of Algebra 10 (1968), 211–230.Google Scholar
12. Mostow, G. D., Fully reducible subgroups of algebraic groups, Amer. J. Math. 78 (1956), 200–221.Google Scholar
13. Santharoubane, L. J., Structure et cohomologie des algèbres de Lie nilpotentes, Thesis (1979), University of Paris 6, France.Google Scholar