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On the Derivation Algebras of Lie Algebras

Published online by Cambridge University Press:  20 November 2018

Shigeaki Tôgô*
Affiliation:
Northwestern University and Hiroshima University
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Let L be a Lie algebra over a field of characteristic 0 and let D(L) be the derivation algebra of L, that is, the Lie algebra of all derivations of L. Then it is natural to ask the following questions: What is the structure of D(L)? What are the relations of the structures of D(L) and L? It is the main purpose of this paper to present some results on D(L) as the answers to these questions in simple cases.

Concerning the questions above, we give an example showing that there exist non-isomorphic Lie algebras whose derivation algebras are isomorphic (Example 3 in § 5). Therefore the structure of a Lie algebra L is not completely determined by the structure of D(L). However, there is still some intimate connection between the structure of D(L) and that of L.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1961

References

1. Chevalley, C., Théorie des groupes de Lie, tome III, Act. Sci. Ind., no. 1226 (Paris, 1055).Google Scholar
2. Dixmier, J., Sous-algèbres de Cartan et decompositions de Levi dans les algebres de Lie, Trans. Roy. Soc. Canada, Series III, Section III, 20 (1956), 17-21.Google Scholar
3. Dixmier, J. Sur les représentations unitaires des groupes de Lie nilpotents III, Can. J. Math., 10 (1958), 321-348.Google Scholar
4. Dixmier, J. and Lister, W. G., Derivations of nilpotent Lie algebras, Proc. Amer. Math. Soc, 8 (1957), 155-158.Google Scholar
5. Hochschild, G., Semi-simple algebras and generalized derivations, Amer. J. Math., 64 (1942), 677-694.Google Scholar
6. Léger, G. and Togo, S., Characteristically nilpotent Lie algebras, Duke Math. J., 26 (1959), 623-628.Google Scholar
7. Togo, S., On the derivations of Lie algebras, J. Sci. Hiroshima Univ., Ser. A, 19 (1955), 71-77.Google Scholar