Stem extensions and stem covers of Leibniz algebras. (English) Zbl 1051.17001

Casas, J. M.; Ladra, M.

Georgian Math. J. 9, No. 4, 659-669 (2002).

The paper concerns central extensions of Leibniz algebras \(\left( \mathfrak{g}\right) :\) \(0\rightarrow \mathfrak{n}\rightarrow \mathfrak{g} \rightarrow \mathfrak{q}\rightarrow 0\). If the mapping \(\theta _{\ast }\left( \mathfrak{g}\right) :HL^{2}\left( \mathfrak{q},\mathfrak{n}\right) \rightarrow \mathfrak{n}\) in the natural exact sequence in Leibniz homology [see J.-L. Loday, Cyclic homology, Berlin: Springer (1992; Zbl 0780.18009), Enseign. Math., II. Sér. 39, No. 3–4, 269-292 (1993; Zbl 0806.55009)] is an epimorphism (or an isomorphism), then \(\left( \mathfrak{g }\right) \) is called a stem extension (a stem cover). The goal of this paper is to obtain some properties of stem extensions and stem covers of Leibniz algebras. For example:
– Every central extension class of a \(K\)-vector space (trivial \(\mathfrak{q }\)-module) \(\mathfrak{n}\) by a Leibniz algebra \(\mathfrak{q}\) is forward induced from a stem extension;
– Every stem extension of \(\mathfrak{q}\) is an epimorphism image of some stem cover.
The final section is devoted to central extensions \(\left( \mathfrak{g} \right) \) in which \(\mathfrak{q}\) is a perfect Leibniz algebra. The authors apply the obtained properties to some examples of universal central extensions (a) of the Lie algebra of matrices with entries in an associative unital algebra whose trace is zero, (b) of derivations of Laurent polynomials, and to others.

Reviewer: Jan Kubarski (Łódź)

MSC:

17A32

Leibniz algebras

Keywords:

Leibniz algebra; Leibniz cohomology; Leibniz homology; central extension; stem extension; stem cover; perfect Leibniz algebra

Citations:

Zbl 0780.18009; Zbl 0806.55009