On stem covers and the universal central extensions of Lie crossed modules

Some Properties of the Schur Multiplier and Stem Covers of Leibniz Crossed Modules

“…In the following, we determine the structure of stem covers of Leibniz crossed module which is analogous to the similar works in group and in Lie crossed modules [30,26]. so (e) is stem cover of (n, q, δ).…”

“…On the other hand, 0 −→ ((a, b, σ) + (h, p, σ) ann )/(h, p, σ) ann −→ (h, p, σ) Lie −→ (T, L, τ ) −→ 0 is a stem extension of (T, L, τ ) in the category XLie of crossed modules in Lie algebras, so by the proof of Theorem 3.6 in [30], there is a surjective homomorphism α = (α 1 , α 2 ) from (M 1 , P 1 , ϑ 1 ) to (h, p, σ) Lie , where (M 1 , P 1 , ϑ 1 ) is a stem cover of (T, L, τ ) in XLie. Now, combining [30,Corollary 3.5] with [28,Proposition 22] It is easy to check that β1 and β2 are isomorphisms and so β is an isomorphism of crossed modules, as required.…”

The second relative homology of Leibniz algebras

“…In continuation of the section, we will construct a relative stem cover for the direct sum of two pairs of Leibniz algebras in terms of given relative stem covers of them, which is a generalization of [29,Corollary 5.6] for Leibniz algebras. To do this, we need the following lemma.…”

On Commutativity Degree of Crossed Modules

“…In 2019, Yavari and Salemkar [18] presented a generalized crossed module and investigated the category of generalized crossed modules. Also, in [2] and [12] the notions of stem cover and universal central extension have been extended for lie crossed modules.…”