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On Lie algebra obstructions

Published online by Cambridge University Press:  17 April 2009

J. Knopfmacher
J. Knopfmacher
Affiliation:
University of the Witwatersrand, Johannesburg, South Africa.
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Abstract

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A basic problem in the theory of Lie algebra extensions concerns a given homomorphism X of a Lie algebra L into the Lie algebra of outer derivations of a Lie algebra B. In analogy with the theory of group extensions, Mori and HochschiId developed the concept of an obstruction to X being the homomorphism defined by some Lie algebra extension of B by L. This note considers an alternative approach to this theory, which is particularly simple when applied to the problem of realizing arbitrary three-cohomology classes of L as obstructions. The approach is analogous to one for groups, which was given recently by Gruenberg.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 1 , Issue 2 , August 1969 , pp. 281 - 288
Copyright
Copyright © Australian Mathematical Society 1969

References

[1]Eilenberg, Samuel and MacLane, Saunders, “Cohomology theory in abstract groups”, Parts I and II, Ann. of Math. 48 (1947), 51–78 and 326–341.CrossRefGoogle Scholar
[2]Gruenberg, K.W., “Resolutions by relations”, J. Lond. Math. Soc. 35 (1960), 481–494.CrossRefGoogle Scholar
[3]Gruenberg, K.W., “A new treatment of group extensions”, Math. Zeitschr. 102 (1967), 340–350.CrossRefGoogle Scholar
[4]Hochschild, G., “Cohomology and representations of associative algebras”, Duke Math. J. 14 (1947), 921–948.CrossRefGoogle Scholar
[5]Hochschild, G., “Cohomology of restricted Lie algebras”, Amer. J. Math. 76 (1954), 555–580.CrossRefGoogle Scholar
[6]Hochschild, G., “Lie algebra kernels and cohomology”, Amer. J. Math. 76 (1954), 698–716.CrossRefGoogle Scholar
[7]Knopfmacher, J., “Extensions in varieties of groups and algebras”, Acta Math. 115 (1966), 17–50.CrossRefGoogle Scholar
[8]Knopfmacher, J., “Some homological formulae”, J. Algebra 9 (1968), 212–219.CrossRefGoogle Scholar
[9]Lue, Abraham S.-T., “Crossed homomorphisms of Lie algebras”, Proc. Cambridge Philos. Soc. 62 (1966), 577–581.CrossRefGoogle Scholar
[10]Mori, Mitsuya, “On the three-dimensional cohomology group of Lie algebras”, J. Math. Soc. Japan 5 (1953), 171–183.CrossRefGoogle Scholar