A non-abelian exterior product and homology of Leibniz algebras

Abstract

Abstract. We introduce a non-abelian exterior product of two crossed modules of Leibniz algebra and investigate its relation to the low dimensional Leibniz homology. Later this non-abelian exterior product is applied to the construction of eight term exact sequence in Leibniz homology. Also its relationship to the universal quadratic functor is established, which is applied to the comparison of the second Lie and Leibniz homologies of a Lie algebra.

The second relative homology of Leibniz algebras

“…where the notation x j means that the variable x j is omitted. It is easily seen that HL ( g) ∼ = g ab and by [14,Theorem 3.16], HL 2 (g) ∼ = ker(g g −→ g). Let n be an ideal of g, then the natural epimorphism π : g → g/n induces a morphism of chain complexes π * : (CL * (g), ∂ * ) → (CL * (g/n), ∂ * ).…”

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

“…Now consider the forgetful functor U 2 : Lb → Set that assigns to a Leibniz algebra q its underlying set. It is well-know (see for instance [12]) that U 2 has as left adjoint the free Leibniz algebra functor F 2 : Set → Lb.…”

“…Remark 3.7. By assumptions of Lemma 3.6, let Ker(ϕ 1 , ϕ 2 ) = (a, b, σ) be, then we have the natural induced map of Leibniz algebras ψ 1 : p a+b m −→ p m and ψ 2 : p b −→ p p, such that Im(ψ 1 ) = Ker(ϕ 2 ϕ 1 ) and Im(ψ 2 ) = Ker(ϕ 2 ϕ 2 ) (see [12]). So we may assume that the Ker(ϕ 2 ϕ 1 ) is an ideal of p m generated by all elements p a, a p, b m and m b for any p ∈ p, a ∈ a, m ∈ m and b ∈ b.…”

“…Moreover, we show the category of Leibniz crossed modules has enough projective objects. In section 3 we describe the Schur multiplier of a Leibniz crossed module and analyze its interplay with the non-abelian exterior product of Leibniz algebras given in [12]. In concrete we show that M(n, q, δ) ∼ = Ker ((q n, q q, id δ) −→ (n, q, δ)) and the existence of the six-term exact sequence…”

“…This structure is not only important by algebraic reasons, but also for its applications in other branches such as Geometry or Physics (see for instance [11,18,20,25] An active research line consists in the extension of properties from Lie algebras to Leibniz algebras. As an example of these generalizations, stem covers and stem extensions of a Leibniz algebras where studied in [10]; in [19] was extended to Leibniz algebras the notion of non-abelian tensor product of Lie algebras introduced by Ellis in [14]; in [12], authors investigated the interplay between the non-abelian tensor and exterior product of Leibniz algebras with the low dimensional Leibniz homology of Leibniz algebras.…”