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D G ] 2 6 Fe b 20 04 EXTENSIONS OF LIE ALGEBRAS

Profile image of Wolfgang RuppertWolfgang Ruppert

2008

Abstract

We review (non-abelian) extensions of a given Lie algebra, identify a 3-dimensional cohomological obstruction to the existence of extensions. A striking analogy to the setting of covariant exterior derivatives, curvature, and the Bianchi identity in differential geometry is spelled out.

Figures (1)
The left hand vertical column describes h as a central extension of ad(h) with abelian kernel Z(H) which is moreover killed under the action of g via @; it is given by a cohomology class [v] € H?(ad(); Z(b))%. In order to get an extension e of g with kernel 6 as in the third row we have to check that the cohomology class |v] is in the image of i* : H?(e9; Z(h)) 4 H?(ad(h); Z(h))9. It would be interesting to express this in terms of of the Hochschild-Serre exact sequence, see [6]. and the following diagram, where the bottom right hand square is a pullback (com- pare with remark 7):
The left hand vertical column describes h as a central extension of ad(h) with abelian kernel Z(H) which is moreover killed under the action of g via @; it is given by a cohomology class [v] € H?(ad(); Z(b))%. In order to get an extension e of g with kernel 6 as in the third row we have to check that the cohomology class |v] is in the image of i* : H?(e9; Z(h)) 4 H?(ad(h); Z(h))9. It would be interesting to express this in terms of of the Hochschild-Serre exact sequence, see [6]. and the following diagram, where the bottom right hand square is a pullback (com- pare with remark 7):

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References (15)

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