Journal of Lie Theory
8(1), 173-182 (1998)

Extensions abéliennes des bigèbres de Lie

M. Benayed

Miloud Benayed
Université des Sciences de Lille I
UFR de Mathématiques Pures
URA 751 associée au CNRS
F-59655 Villeneuve d'Ascq Cedex

Abstract: In this paper we generalize the results about central extensions of Lie bialgebras obtained in [3] to the case of abelian extensions, i.e. to extensions with abelian, but not necessarily central, kernel. An explicit classification of such extensions is given, and the question of the realizability of the classifying space as a (second) cohomology group is adressed. The realizability question is answered in the affirmative in the case of one dimensional kernel, generalizing a result for central extensions given in [4]. For higher dimensional kernel the classifying space does not in general admit a group structure, it may even be empty. However, if this classifying space is not empty, then it can be described by what one may call a non abelian cohomology of Lie bialgebras.

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