Third homology groups of universal central extensions of a Lie algebra. (English) Zbl 1054.17003

Gnedbaye, Allahtan Victor

Afr. Mat., Sér. III 10, 46-63 (1999).

Leibniz algebras form a noncommutative version of Lie algebras and the Chevalley-Eilenberg homology theory \(H_\ast\) is replaced by an homology theory \(HL_\ast\) [J.-L. Loday, Enseign. Math. (2) 39, 269–293 (1993; Zbl 0806.55009)].
The author studies universal central extensions of perfect Leibniz algebras, computes and compares the homology groups \(HL_3({\mathfrak U})\), \(HL_3({\mathfrak u})\) and \(H_3({\mathfrak u})\), where \({\mathfrak U}\) (resp. \({\mathfrak u}\)) is the universal central extension of a perfect Lie algebra \({\mathfrak g}\) in the category of Leibniz (resp. Lie) algebras.

Reviewer: Marek Golasiński (Toruń)

MSC:

17A32	Leibniz algebras
17B55	Homological methods in Lie (super)algebras
18G40	Spectral sequences, hypercohomology

Keywords:

Leibniz algebra, Lie algebra, non-commutative homology, spectral sequence, universal central extension