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Dual Space Derivations and H2(L, F) of Modular Lie Algebras

Published online by Cambridge University Press:  20 November 2018

Rolf Farnsteiner
Rolf Farnsteiner*
Affiliation:
University of California at Riverside, Riverside, California
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It is well-known that the classical vanishing results of the cohomology theory of Lie algebras depend on the characteristic of the underlying base field. The theorems of Cartan and Zassenhaus, for instance, entail that non-modular simple Lie algebras do not admit non-trivial central extensions. In contrast, early results by Block [3] prove that this conclusion loses its validity if the underlying base field has positive characteristic.

Central extensions of a given Lie algebra L, or equivalently its second cohomology group H(L, F), can be conveniently described by means of derivations φ:L → L*.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 39 , Issue 5 , 01 October 1987 , pp. 1078 - 1106
Copyright
Copyright © Canadian Mathematical Society 1987

References

1. Barnes, D. W., On the cohomology of soluble Lie algebras, Math. Z. 101 (1967), 343–349.Google Scholar
2. Block, R. E., Trace forms on Lie algebras, Can. J. Math. 14 (1968), 553–564.Google Scholar
3. Block, R. E., On the extensions of Lie algebras, Can. J. Math. 20 (1968), 1439–1450.Google Scholar
4. Cartan, H. and Eilenberg, S., Homological algebra, (Princeton University Press).Google Scholar
5. Farnsteiner, R., The associative forms of the graded Cartan type Lie algebras, Trans. Amer. Math. Soc. 295 (1986), 417–427.Google Scholar
6. Farnsteiner, R., Central extensions and invariant forms oj graded Lie algebras, Algebras, Groups, Geom. 3 (1986), 431–455.Google Scholar
7. Frank, M. S., A new class of simple Lie algebras, Proc. Nat. Acad. Sci., U.S.A. 40 (1954), 713–714.Google Scholar
8. Hochschild, G. P., Cohomology of restricted Lie algebras, Amer. J. Math. 76 (1954), 555–580.Google Scholar
9. Kostrikin, A. I. and Safarevic, I. R., Graded Lie algebras of finite characteristic, Math. USSR Izv: 3 (1969), 237–304.Google Scholar
10. Seligman, G. B., Modular Lie algebras, Ergebuisse der Mathematikundihrer Grenzgebiete, Band 40 (Springer-Verlag, 1967).CrossRefGoogle Scholar