# zbMATH — the first resource for mathematics

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

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