Journal of Lie Theory 22 (2012), No. 1, 001--015
Copyright Heldermann Verlag 2012
On the Cohomology of Split Lie Algebra Extensions
Dieter Degrijse
Dept. of Mathematics, K.U. Leuven-Kortrijk, Etienne Sabbelaan 53, 8500 Kortrijk, Belgium
dieter.degrijse@kuleuven-kortrijk.be
Nansen Petrosyan
Dept. of Mathematics, K.U. Leuven-Kortrijk, Etienne Sabbelaan 53, 8500 Kortrijk, Belgium
nansen.petrosyan@kuleuven-kortrijk.be
We introduce the notion of compatible actions in the context of split extensions of Lie algebras over a field k. Using compatible actions, we construct new resolutions to compute the cohomology of semi-direct products of Lie algebras and give an alternative way to construct the Hochschild-Serre spectral sequence associated to a split extension. Finally, we describe several instances in which this spectral sequence collapses at the second page and obtain a sharper bound for its length in the finite dimensional case.
Keywords: Lie algebra cohomology, free resolutions, Hochschild-Serre spectral sequence.
MSC: 17B56, 18G60, 18G40
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