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Characteristically Nilpotent Algebras

Published online by Cambridge University Press:  20 November 2018

T. S. Ravisankar
T. S. Ravisankar*
Affiliation:
The Ramanujan Institute, University of Madras, Madras, India Birla Institute of Technology and Science, Pilani, India
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Our aim in this paper is to extend (Theorem 1.7) to general algebras a classical result of Lie algebras due to Léger and Togo [6]. This extension requires, in turn, extension to general algebras of the concept of characteristically nilpotent algebras introduced by Dixmier and Lister [3] for Lie algebras. Based on this extended concept, we introduce in § 2 a new concept of radical (and semisimplicity) for general algebras and Lie triple systems. We study in some detail the consequences of the newly introduced concepts, furnishing necessary examples. With a stronger notion of characteristically nilpotent Mal'cev algebra arising out of these concepts, we obtain (Proposition 3.6) for such an algebra the parallel to the Leger-Tôgô result mentioned at the outset. In § 4, we deal with a further generalization of the concept of characteristic nilpotency leading to extension of very recent results of Chao [1] and Tôgô [12].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 2 , April 1971 , pp. 222 - 235
Copyright
Copyright © Canadian Mathematical Society 1971

References

1. Chao, C., A non-imbedding theorem of nilpotent Lie algebras, Pacific J. Math. 22 (1967), 231–234.Google Scholar
2. Chao, C., Some characterizations of nilpotent Lie algebras, Math. Z. 108 (1968), 40–42.Google Scholar
3. Dixmier, J. and Lister, W. G., Derivations of nilpotent Lie algebras, Proc. Amer. Math. Soc. 8 (1957), 155–158.Google Scholar
4. Jacobson, N., Lie algebras (Interscience, New York, 1962).Google Scholar
5. Jacobson, N., Abstract derivations and Lie algebras, Trans. Amer. Math. Soc. 42 (1937), 206–224.Google Scholar
6. Leger, G. and Togo, S., Characteristically nilpotent Lie algebras, Duke Math. J. 26 (1959), 623–628.Google Scholar
7. Loos, O., Uber eine beziehung zwischen Malcev-algebren und Lie-tripelsystemen, Pacific J. Math. 18 (1966), 553–562.Google Scholar
8. Schafer, R. D., An introduction to nonas so dative algebras (Academic Press, New York, 1966).Google Scholar
9. Seligman, G. B., Characteristic ideals and the structure of Lie algebras, Proc. Amer. Math. Soc. 8 (1957), 159–164.Google Scholar
10. Tôgô, S., On the derivation algebras of Lie algebras, Can. J. Math. 13 (1961), 201–216.Google Scholar
11. Tôgô, S., Lie algebras which have few derivations, J. Sci. Hiroshima Univ. Ser. A 29 (1965), 29–41.Google Scholar
12. Tôgô, S., A theorem on characteristically nilpotent algebras, J. Sci. Hiroshima Univ. Ser. A 33 (1969), 209–212.Google Scholar