Georgian Mathematical Journal Volume 9 (2002), Number 4, 659–669

STEM EXTENSIONS AND STEM COVERS OF LEIBNIZ ALGEBRAS

J. M. CASAS AND M. LADRA

Dedicated to Hvedri Inassaridze on his 70th birthday

Abstract. Some properties on stem extensions and stem covers of Leibniz algebras are studied by means of exact sequences associated to a central extension of Leibniz algebras. We obtain particular results in case of perfect Leibniz algebras.

2000 Mathematics Subject Classification: 17A32.

Key words and phrases: Leibniz algebra, Leibniz (co)homology, central extension, stem extension, stem cover, perfect Leibniz algebra.

1. Introduction

Leibniz algebras, introduced by Loday [12, 13], are a non skew-symmetric generalization of Lie algebras. In brief, a (right) Leibniz algebra over a field K is a K-vector space 𝔤 equipped with a bilinear map called the Leibniz bracket, [−, −] : 𝔤 × 𝔤 → 𝔤 satisfying the (right) Leibniz identity [x, [y, z]] = [[x, y], z] − [[x, z], y], for all x, y, z ∈ 𝔤. If we assume that [x, x] = 0 for all x ∈ 𝔤, then 𝔤 is a K-Lie algebra and the Leibniz identity is the Jacobi identity.

Let (𝔤) : 0 → 𝔫 →^{χ} 𝔤 →^{π} 𝔮 → 0 be an extension of Leibniz algebras. If (𝔤) is a central extension, that is, [n, g] = 0 = [g, n], n ∈ 𝔫, g ∈ 𝔤, then there exists the following natural exact sequence in Leibniz homology [13, 14] associated to (𝔤):

\begin{matrix}HL_{3}(𝔤) → HL_{3}(𝔮) & → Coker(τ) → HL_{2}(𝔤) → HL_{2}(𝔮) → 𝔫 \\ → & HL_{1}(𝔤) → HL_{1}(𝔮) → 0, \end{matrix}  (1)

where the map τ : 𝔫 ⊗ 𝔫 → HL_{1}(𝔤) ⊗ 𝔫 ⊕ 𝔫 ⊗ HL_{1}(𝔤) is given by τ(a ⊗ b) = (d(χ(a)) ⊗ b, −a ⊗ d(χ(b))) and d : 𝔤 → HL_{1}(𝔤) = 𝔤/[𝔤,𝔤] is the natural projection [4]. On the other hand, we have the natural isomorphism θ_{⋆} : HL^{2}(𝔮, 𝔫) →^{∼} Hom(HL_{2}(𝔮), 𝔫) obtained from Theorem 5.3 in [3] when 𝔫 is a trivial 𝔮-module.

If the morphism θ_{⋆}(𝔤) : HL_{2}(𝔮) → 𝔫 in the sequence (1) is an epimorphism or an isomorphism, then (𝔤) is called a stem extension (i.e., 𝔫 ⊆ [𝔤, 𝔤]) and a stem cover (i.e., HL_{1}(𝔤) →^{∼} HL_{1}(𝔮) and π_{⋆} : HL_{2}(𝔤) → HL_{2}(𝔮) is the zero map), respectively (see [2] for the characterization of these central extensions). These extensions in the category of groups were studied in [6, 9] and in the category of Lie algebras in [1, 8].

The goal of this paper is to use the sequence (1) and the isomorphism θ_{⋆} together with the sequence (2) (see below) in order to obtain some properties of stem extensions and stem covers of Leibniz algebras. In the final section we

ISSN 1072-947X / $8.00 / \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag www.heldermann.de

consider the particular case of central extensions (𝔤) in which 𝔮 is a perfect Leibniz algebra (i.e., HL_{1}(𝔮) = 0).

2. HL^{2} and Extensions

Let (𝔤): 0 → 𝔫 →^{χ} 𝔤 →^{π} 𝔮 → 0 be an abelian extension of Leibniz algebras [3], then (𝔤) induces a 𝔮-module (representation [13, 14]) structure on 𝔫 [3, 13, 14]. When 𝔫 is a (right) 𝔮-module, then it is said that (𝔤) is a 𝔮-extension of 𝔫 if the 𝔮-module structure induced by (𝔤) coincides with the previous one [3]. Two extensions (𝔤) and (𝔤′) of 𝔮 by 𝔫 are congruent if there exists an isomorphism of Leibniz algebras f : 𝔤 → 𝔤′ which induces identities on 𝔫 and 𝔮. We denote by Ext(𝔮, 𝔫) the set of congruence classes (𝔤) of 𝔮-extensions of 𝔫.

We know that HL^{2}(𝔮, 𝔫) ≅ Ext(𝔮, 𝔫) [14] and that associated to the 𝔮extension (𝔤) there exists the following natural exact sequence [3]:

0 → Der(𝔮, 𝔫) →^{Der(π)} Der(𝔤, 𝔫) →^{ρ} Hom_{𝔮}(𝔫, 𝔫) →^{θ^{⋆}(𝔤)} HL^{2}(𝔮, 𝔫) →^{π^{⋆}} HL^{2}(𝔤, 𝔫). (2)

We define an application Δ : Ext(𝔮, 𝔫) → HL^{2}(𝔮, 𝔫), Δ[𝔤] = ξ = θ^{⋆}(𝔤)(1_{𝔫}), which is well defined by naturality of the sequence (2). Moreover we can see that Δ is bijective (it suffices to use the proof of Theorem 3.3, p. 207 in [10]). If Ext(𝔮, 𝔫) is endowed with the structure of a K-vector space by means of the Baer sum [3], then Δ is a linear map; so Δ is a linear isomorphism between Ext(𝔮, 𝔫) and HL^{2}(𝔮, 𝔫) and we can see that the natural structure of Ext(𝔮, 𝔫) induced by Δ from the vector space structure of HL^{2}(𝔮, 𝔫) is precisely the structure given in [3]. It is also known that Ext(𝔮, −) is a covariant functor from the category of (right) 𝔮-modules to the category of K-vector spaces. Obviously, Δ maps the neutral element of Ext(𝔮, 𝔫), which is the class of split extensions, to the neutral element of HL^{2}(𝔮, 𝔫). If 𝔫 is a trivial 𝔮-module, then Ext(𝔮, 𝔫) is the set of congruence classes of central extensions of 𝔮 by 𝔫.

Finally, by Lemma 5.2 in [3], the following identities hold:

θ_{⋆}Δ[𝔤] = θ_{⋆}θ^{⋆}(𝔤)(1_{𝔫}) = θ_{⋆}(𝔤).  (3)

3. Morphisms of Extensions with Abelian Kernel

In this section we develop some technical results on a morphism of extensions with abelian kernel which are used in the next sections.

Let (𝔤): 0 → 𝔫 →^{χ} 𝔤 →^{π} 𝔮 → 0 be a 𝔮-extension of 𝔫 and let α : 𝔫 → 𝔫′ be a homomorphism of 𝔮-modules, that is, α[q, n] = [q, α(n)], α[n, q] = [α(n), q], n ∈ 𝔫, q ∈ 𝔮, and let (𝔤′): 0 → 𝔫′ →^{χ′} 𝔤′ →^{π′} 𝔮 → 0 be a 𝔮-extension with Δ[𝔤′] = ξ′ ∈ HL^{2}(𝔮, 𝔫′).

Proposition 1. There exists a homomorphism of Leibniz algebras f : 𝔤 → 𝔤′ such that the diagram

[figure: 0; 𝔫; 𝔤; 𝔮; 0; 0; 𝔫′; 𝔤′; 𝔮; 0; (𝔤) :; (𝔤′) :; χ; π; χ′; π′; α; f]

is commutative if and only if α_{⋆}(ξ) = ξ′ ∈ HL^{2}(𝔮, 𝔫′).

Proof. The naturality in (2) implies α_{⋆}(ξ)=α_{⋆}θ^{⋆}(𝔤)(1_{𝔫}) = θ^{⋆}(𝔤)α_{⋆}1_{𝔫} = θ^{⋆}(𝔤)α = θ^{⋆}(𝔤)α^{⋆}(1_{𝔫′}) = θ^{⋆}(𝔤′)(1_{𝔫′}) = ξ′.

Conversely, for the 𝔮-extension (𝔤) we construct the forward induced extension (^{α}𝔤) [3], obtaining the morphism of extensions (α, f_{α}, 1) : (𝔤) → (^{α}𝔤). Thus α_{⋆}(ξ) = α_{⋆}θ^{⋆}(𝔤)(1_{𝔫}) = α_{⋆}Δ[𝔤] = Δ[^{α}𝔤]. Consequently Δ[^{α}𝔤] = α_{⋆}(ξ) = ξ′ = Δ[𝔤′] and then (^{α}𝔤) ≡ (𝔤′); thus f′f_{α} : 𝔤 → ^{α}𝔤 → 𝔤′ verifies the Proposition. □

We note that the composition f′f_{α} is not unique; in fact there are many maps f′f_{α} with the required properties since the previous result is a particular case of Theorem 3.1 in [3], where it is confirmed that there are as many homomorphisms f′f_{α} as derivations from 𝔮 to 𝔫′.

Let γ : 𝔮̄ → 𝔮 be a homomorphism and let (\overline{𝔤}): 0 → 𝔫 →^{\overline{χ}} \overline{𝔤} →^{\overline{π}} \overline{𝔮} → 0 be a 𝔮̄-extension with Δ[𝔤̄] = ξ̄ ∈ HL^{2}(𝔮̄, 𝔫).

Proposition 2. There exists f̄: 𝔤̄ → 𝔤 such that the diagram

[figure: 0; 𝔫; 𝔤̄; 𝔮̄; 0; 0; 𝔫; 𝔤; 𝔮; 0; (𝔤̄) :; (𝔤) :; γ; f̄]

is commutative if and only if HL^{2}(γ, 𝔫)(ξ) = γ^{⋆}(ξ) = ξ̄.

Proof. If there exists f̄, then naturality of (2) implies that ξ̄ = θ^{⋆}(𝔤̄)(1_{𝔫}) = HL^{2}(γ, 𝔫)θ^{⋆}(𝔤)(1_{𝔫}) = γ^{⋆}(ξ).

Conversely, from the 𝔮-extension (𝔤) we construct the backward induced extension (𝔤_{γ}) [3] and so we have the morphism of extensions (1, γ̄, γ) : (𝔤_{γ}) → (𝔤). From here, Δ[𝔤_{γ}] = γ^{⋆}Δ[𝔤] = γ^{⋆}(ξ) = ξ̄ = Δ[𝔤̄]; consequently (𝔤_{γ}) ≡ (𝔤̄) and then f̄: 𝔤̄ →^{∼} 𝔤_{γ} → 𝔤 satisfies the Proposition. □

Proposition 3. Given the following diagram

[figure: 0; 𝔫; 𝔤̄; 𝔮̄; 0; 0; 𝔫′; 𝔤′; 𝔮; 0; (𝔤̄) :; (𝔤′) :; γ; α]

there exists f : 𝔤̄ → 𝔤′ making the diagram commutative if and only if α_{⋆}(Δ[𝔤̄]) = γ^{⋆}(Δ[𝔤′]). Moreover, the solution set is in the one-to-one correspondence with the derivation set of 𝔮̄ to 𝔫′.

Proof. If f exists, we consider the composition (1, γ̄, γ)(α, σ, 1) : (𝔤̄) → (𝔤′_{γ}) → (𝔤′), then Propositions 1 and 2 imply that α_{⋆}(Δ[𝔤̄]) = Δ[𝔤′_{γ}] = γ^{⋆}(Δ[𝔤′]).

Conversely, we consider the composition (1, γ̄, γ)(1, −, 1)(α, σ, 1) : (𝔤̄) → (^{α}𝔤̄) → (𝔤′_{γ}) → (𝔤′) and, applying Propositions 1 and 2, we have that Δ[^{α}𝔤̄] = α_{⋆}(Δ[𝔤̄]) = γ^{⋆}(Δ[𝔤′]) = Δ[𝔤′_{γ}]; consequently, (^{α}𝔤̄) and (𝔤′_{γ}) are congruent and (1, −, 1) = (1, ϕ, 1) is the wanted morphism.

A proof of the second statement in this Proposition can be seen in Theorem 3.1 in [3]. □

4. Stem Extensions and Stem Covers

Definition 1 ([2]). A central extension of Leibniz algebras (𝔤) : 0 → 𝔫 → 𝔤 → 𝔮 → 0 is called a stem extension if 𝔫 ⊆ [𝔤, 𝔤]. It is said that (𝔤) is a stem cover if 𝔤_{ab} ≅ 𝔮_{ab} and HL_{2}(𝔤) → HL_{2}(𝔮) is the zero map.

Proposition 4. Every central extension class of a K-vector space (trivial 𝔮-module) 𝔫 by a Leibniz algebra 𝔮 is forward induced from a stem extension.

Proof. Choose any central extension class (𝔤) : 0 → 𝔫 → 𝔤 → 𝔮 → 0, then θ_{⋆}(𝔤) : HL_{2}(𝔮) → 𝔫 factors as iτ : HL_{2}(𝔮) → 𝔫 ∩ [𝔤, 𝔤] = 𝔫_{1} → 𝔫. As 𝔫_{1} is a trivial 𝔮-module, then, given τ, there exists a central extension (𝔤_{1}) ∈ HL^{2}(𝔮, 𝔫_{1}) such that θ_{⋆}(𝔤_{1}) = τ. Moreover, (𝔤_{1}) is a stem extension. By naturality of the sequence (1) on the forward construction (𝔤_{1}) → i_{⋆}(𝔤_{1}) [3], we have that θ_{⋆}i_{⋆}(𝔤_{1}) = iτ = θ_{⋆}(𝔤), i.e., i_{⋆}(𝔤_{1}) = (𝔤), and so (𝔤) is forward induced by (𝔤_{1}), which is a stem extension. □

Proposition 5. Let 𝔮 be a Leibniz algebra and let 𝔲 be a subspace of HL_{2}(𝔮), then there exists a stem extension (𝔤) with 𝔲 = Ker θ_{⋆}Δ[𝔤].

Proof. We consider the quotient vector space 𝔫 = HL_{2}(𝔮)/𝔲 as a trivial 𝔮module. We consider the central extension (𝔤) : 0 → 𝔫 → 𝔤 → 𝔮 → 0 ∈ HL^{2}(𝔮, 𝔫). Thus θ_{⋆}Δ[𝔤] = θ_{⋆}(𝔤) ∈ Hom(HL_{2}(𝔮), 𝔫). If θ_{⋆}(𝔤) : HL_{2}(𝔮) → 𝔫 = HL_{2}(𝔮)/𝔲 is the canonical projection; then there exists a central extension (𝔤) : 0 → 𝔫 → 𝔤 → 𝔮 → 0 such that θ_{⋆}Δ[𝔤] = θ_{⋆}(𝔤) is the canonical projection. Associated to (𝔤), we have the exact sequence (1), in which 𝔲 = Ker θ_{⋆}(𝔤) = Ker θ_{⋆}Δ[𝔤]. Moreover, (𝔤) is a stem extension since θ_{⋆}Δ[𝔤] = θ_{⋆}(𝔤) is an epimorphism. □

Remark 1. A stem extension is a stem cover if and only if 𝔲 = 0.

Remark 2. Any stem cover (𝔤) : 0 → 𝔫 → 𝔤 → 𝔮 → 0 is isomorphic to a stem cover (𝔤′) : 0 → HL_{2}(𝔮) → 𝔤′ → 𝔮 → 0 with θ_{⋆}Δ[𝔤′] = 1_{HL_{2}(𝔮)}. Indeed, there always exists 𝔤′ and thus it suffices to take 𝔲 = 0 in Proposition 5; if φ : 𝔫 → HL_{2}(𝔮) is the inverse of θ_{⋆}Δ[𝔤], then the naturality of the isomorphism θ_{⋆} : HL^{2}(𝔮, 𝔫) →^{∼} Hom(HL_{2}(𝔮), 𝔫) implies θ_{⋆}φ_{⋆}(Δ[𝔤]) = φ_{⋆}θ_{⋆}(Δ[𝔤]) = φθ_{⋆}(Δ[𝔤]) = 1_{HL_{2}(𝔮)} so that we can choose (𝔤′) such that Δ[𝔤′] = φ_{⋆}Δ[𝔤]. By Proposition 1 there exists f : 𝔤 → 𝔤′ making the following diagram commutative:

[figure: 0; 𝔫; 𝔤; 𝔮; 0; 0; HL_{2}(𝔮); 𝔤′; 𝔮; 0; (𝔤) :; (𝔤′) :; f; φ]

The name of stem cover is motivated by the following result.

Proposition 6. Every stem extension of 𝔮 is an epimorphic image of some stem cover.

Proof. Let (𝔤) : 0 → 𝔫 → 𝔤 → 𝔮 → 0 be a stem extension characterized by Δ[𝔤] = ξ ∈ HL^{2}(𝔮, 𝔫); then φ = θ_{⋆}(ξ) = θ_{⋆}(Δ[𝔤]) = θ_{⋆}(𝔤) : HL_{2}(𝔮) → 𝔫 is an epimorphism. In order to prove the Proposition, we must find η ∈ HL^{2}(𝔮, HL_{2}(𝔮)) with φ_{⋆}(η) = ξ and θ_{⋆}(η) = 1_{HL_{2}(𝔮)} (i.e., η is an element in the second cohomology K-vector space which characterizes a stem cover), where φ_{⋆} is the induced morphism by naturality of the isomorphism θ_{⋆} on φ.

Let η ∈ HL^{2}(𝔮, HL_{2}(𝔮)) be such that θ_{⋆}(η) = 1_{HL_{2}(𝔮)}; then θ_{⋆}(ξ − φ_{⋆}(η)) = φ − φ_{⋆}θ_{⋆}(η) = 0; consequently φ_{⋆}(η) = ξ. Obviously, η verifies the required conditions. Now, let (𝔤′) : 0 → HL_{2}(𝔮) → 𝔤′ → 𝔮 → 0 ∈ HL^{2}(𝔮, HL_{2}(𝔮)) be such that Δ[𝔤′] = η; by Proposition 1 there exists f : 𝔤′ → 𝔤 such that (φ, f, 1) : (𝔤′) → (𝔤) is an epimorphism. □

Proposition 7. There exists a unique isomorphism class of stem covers of 𝔮.

Proof. By Remark 2, stem covers are of the form (𝔤) : 0 → HL_{2}(𝔮) → 𝔤 → 𝔮 → 0 such that θ_{⋆}Δ[𝔤] = 1_{HL_{2}(𝔮)}. Fix a stem cover (𝔤); for another stem cover (𝔤′) we have that θ_{⋆}Δ[𝔤] = θ_{⋆}Δ[𝔤′] = 1_{HL_{2}(𝔮)}; then [𝔤] = [𝔤′]. □

In this case we shall speak of the stem cover of 𝔮.

Proposition 8. Let (𝔤̄) : 0 → HL_{2}(𝔮̄) → 𝔤̄ → 𝔮̄ → 0 be a stem cover and let (𝔤) : 0 → 𝔫 → 𝔤 → 𝔮 → 0 be a central extension. Then every homomorphism f : 𝔮̄ → 𝔮 can be lifted to a map f′ : 𝔤̄ → 𝔤.

Proof. Let Δ[𝔤] = ξ ∈ HL^{2}(𝔮, 𝔫). We define φ = f^{⋆}θ_{⋆}(ξ) : HL_{2}(𝔮̄) → 𝔫. Since η = Δ[𝔤̄] ∈ HL^{2}(𝔮̄, HL_{2}(𝔮̄)) is a stem cover with θ_{⋆}(η) = θ_{⋆}Δ[𝔤̄] = θ_{⋆}(𝔤̄) = 1_{HL_{2}(𝔮̄)}, we have θ_{⋆}φ_{⋆}(η) = φ_{⋆}θ_{⋆}(η) = φ = f^{⋆}θ_{⋆}(ξ) = θ_{⋆}f^{⋆}(ξ) (by the naturality in the isomorphism θ_{⋆}), and so φ_{⋆}(η) = f^{⋆}(ξ), i.e., φ_{⋆}(Δ[𝔤]) = f^{⋆}(Δ[𝔤]); Proposition 3 ends the proof. □

Proposition 9. If (𝔤) : 0 → 𝔫 → 𝔤 → 𝔮 → 0 is a stem extension, then the following sequence is exact:

HL_{3}(𝔤) → HL_{3}(𝔮) → HL_{1}(𝔤)⊗𝔫⊕𝔫⊗HL_{1}(𝔤) → HL_{2}(𝔤) → HL_{2}(𝔮) → 𝔫 → 0

Proof. HL_{2}(𝔮) → 𝔫 is an epimorphism and see Remark 1 in [4]. □

Corollary 1. Let (𝔤) : 0 → HL_{2}(𝔮) → 𝔤 → 𝔮 → 0 be a stem cover associated to 𝔮. Then the following sequence is exact:

HL_{3}(𝔤) → HL_{3}(𝔮) → HL_{1}(𝔤) ⊗ HL_{2}(𝔮) ⊕ HL_{2}(𝔮) ⊗ HL_{1}(𝔤) → HL_{2}(𝔤) → 0.

5. Central Extensions of Perfect Leibniz Algebras

Proposition 10. Let (𝔤) : 0 → 𝔫 →^{χ} 𝔤 →^{π} 𝔮 → 0 and (𝔤′) : 0 → 𝔫′ →^{χ′} 𝔤′ →^{π′} 𝔮′ → 0 be central extensions. Let ρ : 𝔫 → 𝔫′ and σ : 𝔮 → 𝔮′ be homomorphisms of Leibniz algebras. Then: i) There exists τ : 𝔤 → 𝔤′ inducing ρ and σ if and only if the following

diagram is commutative:

[figure: HL_{2}(𝔮); 𝔫; HL_{2}(𝔮′); 𝔫′; ρ; σ_{⋆}; θ_{⋆}(𝔤); θ_{⋆}(𝔤′)]

ii) If τ exists, it is unique if and only if Hom(HL_{1}(𝔮), 𝔫′) = 0.

Proof. i) If τ exists, the commutativity of the square follows from naturality of the sequence (1).

Conversely, we consider the following diagram produced by the naturality of the isomorphism θ_{⋆}:

[figure: HL^{2}(𝔮, 𝔫); Hom(HL_{2}(𝔮), 𝔫); HL^{2}(𝔮, 𝔫′); Hom(HL_{2}(𝔮), 𝔫′); HL^{2}(𝔮′, 𝔫′); Hom(HL_{2}(𝔮′), 𝔫′); ∼; ∼; ∼; ρ_{⋆}; ρ_{⋆}; σ^{⋆}; σ^{⋆}; θ′′_{⋆} :; θ_{⋆} :; θ′_{⋆} :]

Let ξ = Δ[𝔤] and ξ′ = Δ[𝔤′], then ρ_{⋆}θ_{⋆}(ξ) = ρ_{⋆}θ_{⋆}(Δ[𝔤]) = ρ_{⋆}θ_{⋆}(𝔤) = ρθ_{⋆}(𝔤) = θ_{⋆}(𝔤′)σ_{⋆} = σ^{⋆}θ_{⋆}(𝔤′) = σ^{⋆}θ′_{⋆}(Δ[𝔤′]) = σ^{⋆}θ_{⋆}(ξ′). So θ′′_{⋆}ρ_{⋆}(ξ) = ρ_{⋆}θ_{⋆}(ξ) = σ^{⋆}θ′_{⋆}(ξ′) = θ′′_{⋆}σ^{⋆}(ξ′), and consequently ρ_{⋆}(ξ) = σ^{⋆}(ξ′). Now Proposition 3 provides τ : 𝔤 → 𝔤′ inducing ρ and σ.

ii) Suppose that there exists τ : 𝔤 → 𝔤′ inducing ρ and σ and let τ′ : 𝔤 → 𝔤′ be another homomorphism inducing ρ and σ, then there are unique homomorphisms f : 𝔤 → 𝔫′ such that τ′ − τ = χ′f and φ : 𝔮 → 𝔫′ such that φπ = f; consequently, τ′ = τ + χ′φπ; that is, for another homomorphism τ′ : 𝔤 → 𝔤′ there exists a unique homomorphism φ : 𝔮 → 𝔫′ such that τ′ = τ + χ′φπ. Conversely, if φ : 𝔮 → 𝔫′ is a homomorphism, then τ′ = τ + χ′φπ induces ρ and σ. τ is unique if and only if τ − τ′ = 0, that is, χ′φπ = 0, which is equivalent to φ ∈ Hom_{Leib}(𝔮, 𝔫′) = 0 and then Hom(HL_{1}(𝔮), 𝔫′) = 0. □

Corollary 2. Under the hypothesis of Proposition 10, if the map τ : 𝔤 → 𝔤′ exists, then it is unique when 𝔮 is a perfect Leibniz algebra (𝔮_{ab} = 0).

Proof. If 𝔮 is a perfect Leibniz algebra, then Hom(HL_{1}(𝔮), 𝔫′) = 0. □

Proposition 11. The isomorphism classes of stem extensions of 𝔮 are in the one-to-one correspondence with the subspaces of HL_{2}(𝔮). Moreover, if 𝔲 and 𝔳 are two subspaces of HL_{2}(𝔮), then 𝔲 ⊆ 𝔳 if and only if there is a map (necessarily surjective) from the stem extension corresponding to 𝔲 to the stem extension corresponding to 𝔳.

Proof. Let (𝔤) : 0 → 𝔫 → 𝔤 → 𝔮 → 0 be a stem extension. Associate to (𝔤) the subspace 𝔲 = Ker θ_{⋆}(𝔤) = Ker(θ_{⋆}(Δ[𝔤]) : HL_{2}(𝔮) → 𝔫). It is clear that the isomorphic stem extension yields the same subspace of HL_{2}(𝔮).

Conversely, let 𝔲 ⊆ HL_{2}(𝔮) be given, 𝔫 = HL_{2}(𝔮)/𝔲 and consider the canonical projection τ : HL_{2}(𝔮) → 𝔫, then there exists an element Δ[𝔤] ∈ HL^{2}(𝔮, 𝔫) such that θ_{⋆}Δ[𝔤] = τ. Obviously, (𝔤) is unique, θ_{⋆}Δ[𝔤] is an epimorphism and then (𝔤) is a stem extension.

Finally, if (𝔤′) : 0 → 𝔫′ → 𝔤′ → 𝔮 → 0 is another stem extension associated to 𝔲, then there exists an isomorphism r : 𝔫 → 𝔫′ such that the following diagram is commutative:

[figure: 0; 𝔲; HL_{2}(𝔮); 𝔫; 0; 0; 𝔲; HL_{2}(𝔮); 𝔫′; 0; θ_{⋆}(𝔤); θ_{⋆}(𝔤′); r]

By Proposition 10 i) there exists σ : 𝔤 → 𝔤′ inducing r : 𝔫 → 𝔫′ and 1_{𝔮} : 𝔮 → 𝔮 such that (r, σ, 1_{𝔮}) : (𝔤) → (𝔤′) is a morphism of extensions; moreover, σ is an isomorphism and then (𝔤) and (𝔤′) are in the same isomorphism class.

For the second statement, we consider a morphism of stem extensions (r, t, 1) : (𝔤) → (𝔤′). Naturality of the sequence (1) implies 𝔲 = Ker θ_{⋆}(𝔤) ⊆ Ker θ_{⋆}(𝔤′) = 𝔳.

Conversely, we first recall that every stem extension is isomorphic to an extension (𝔤) with the canonical projection θ_{⋆}Δ[𝔤]. It is thus enough to consider those. Let 𝔲 ⊆ 𝔳 ⊆ HL_{2}(𝔮), 𝔫 = HL_{2}(𝔮)/𝔲 and 𝔫′ = HL_{2}(𝔮)/𝔳. There exists an epimorphism r : 𝔫 → 𝔫′ such that

[figure: 0; 𝔲; HL_{2}(𝔮); 𝔫; 0; 0; 𝔳; HL_{2}(𝔮); 𝔫′; 0; τ; σ; r]

commutes. Now, if (𝔤) : 0 → 𝔫 → 𝔤 → 𝔮 → 0 is an extension with θ_{⋆}Δ[𝔤] = θ_{⋆}(𝔤) = τ and (𝔤′) : 0 → 𝔫′ → 𝔤′ → 𝔮 → 0 is another extension with θ_{⋆}Δ[𝔤′] = θ_{⋆}(𝔤′) = σ, then by Proposition 10 i) there exists t : 𝔤 → 𝔤′ inducing r and 1; moreover, t is surjective. □

We recall that if 𝔮 is perfect, then Proposition 10 implies that t is uniquely determined by r.

Proposition 12. Let 𝔮 be a perfect Leibniz algebra and let (𝔤) : 0 → 𝔫 → 𝔤 →^{π} 𝔮 → 0 be a stem extension, then the sequence

0 → HL_{2}(𝔤) → HL_{2}(𝔮) →^{θ_{⋆}(𝔤)} 𝔫 → 0

is exact and π_{⋆} : HL_{3}(𝔤) → HL_{3}(𝔮) is an epimorphism.

Proof. Follows from Proposition 9 and keeping in mind that the hypothesis involves HL_{1}(𝔤) = 0. □

Note that from Propositions 11 and 12 it follows that if 𝔮 is a perfect Leibniz algebra, then the second Leibniz homology K-spaces with trivial coefficients of the stem extension of 𝔮 are precisely the subspaces of HL_{2}(𝔮).

Corollary 3. Let 𝔮 be a perfect Leibniz algebra and let (𝔤) : 0 → 𝔫 → 𝔤 → 𝔮 → 0 be a central extension. Then (𝔤) is a stem cover if and only if HL_{1}(𝔤) = HL_{2}(𝔤) = 0.

Proof. From the exact sequence (1) associated to (𝔤), with HL_{1}(𝔤) = HL_{2}(𝔤) = 0, we easily derive that θ_{⋆}(𝔤) is an isomorphism. □

Definition 2. Let 𝔥 and 𝔫 be two-sided ideals of the Leibniz algebra 𝔤. The commutator subalgebra of 𝔥 and 𝔫 is defined as the Leibniz subalgebra of 𝔤 spanned by the brackets [h, n] and [n, h], for all h ∈ 𝔥, n ∈ 𝔫, that is,

[𝔥, 𝔫] = ⟨{[h, n], [n, h] | h ∈ 𝔥, n ∈ 𝔫}⟩.

It is easy to verify that [𝔥, 𝔫] is a two-sided ideal of 𝔤 and that [𝔥, 𝔫] ⊆ 𝔥 ∩ 𝔫. Remark 3. Let (𝔤) : 0 → 𝔫 → 𝔤 → 𝔮 → 0 be a stem extension with

HL_{2}(𝔤) = 0, then (𝔤) is a stem cover. Corollary 3 shows that the converse is true if, in addition, 𝔮 is a perfect Leibniz algebra. In general, however, there are stem covers with HL_{2}(𝔤) different from 0. For example, let 𝔣 be a non-abelian or non-nilpotent free Leibniz algebra and let us consider the lower central series

𝔣_{0} = 𝔣, 𝔣_{n+1} = [𝔣, 𝔣_{n}].

Then the sequence 0 → 𝔣_{n}/𝔣_{n+1} → 𝔣/𝔣_{n+1} → 𝔣/𝔣_{n} → 0 is central for n ≥ 2; moreover, it is a stem cover since (𝔣/𝔣_{n+1})_{ab} ≅ 𝔣_{ab} ≅ (𝔣/𝔣_{n})_{ab} and, on the other hand, the map HL_{2}(𝔣/𝔣_{n+1}) = 𝔣_{n+1}/𝔣_{n+2} → HL_{2}(𝔣/𝔣_{n}) = 𝔣_{n}/𝔣_{n+1} is trivial. Moreover, HL_{2}(𝔣/𝔣_{n+1}) = 𝔣_{n+1}/𝔣_{n+2} is different from 0.

The following Proposition gives, for a perfect Leibniz algebra 𝔮, a description of the stem covers of 𝔮 in terms of free presentations of 𝔮.

Proposition 13. Let 𝔮 be a perfect Leibniz algebra and let 0 → 𝔯 → 𝔣 →^{f} 𝔮 → 0 be a free presentation. Then

0 → HL_{2}(𝔮) → \frac{[𝔣, 𝔣]}{[𝔣, 𝔯]} →^{φ} 𝔮 → 0

is a stem cover of 𝔮, where φ is induced by f.

Proof.

0 → HL_{2}(𝔮) → \frac{[𝔣, 𝔣]}{[𝔣, 𝔯]} →^{φ} 𝔮 → 0

is the universal central extension of 𝔮. Moreover, it is a stem cover since HL_{2}([𝔣, 𝔣]/[𝔣, 𝔯]) = HL_{1}([𝔣, 𝔣]/[𝔣, 𝔯]) = 0 (see Proposition 4.2 in [14]). □

From Proposition 7, when 𝔮 is a perfect Leibniz algebra, we have that any stem cover is isomorphic to 0 → HL_{2}(𝔮) → [𝔣,𝔣]/[𝔣,𝔯] → 𝔮 → 0.

Proposition 14. Let (𝔤) : 0 → 𝔫 →^{χ} 𝔤 →^{π} 𝔮 → 0 be a central extension and let f : 𝔵 → 𝔮 be a homomorphism of Leibniz algebras, where 𝔵 is a perfect Leibniz algebra. Then there exists φ : 𝔵 → 𝔤 such that πφ = f if and only if f_{⋆}(HL_{2}(𝔵)) ⊆ π_{⋆}(HL_{2}(𝔤)). If φ exists, it is uniquely determined.

Proof. If φ exists, then the functor HL_{2}(−) preserves the composition so that f_{⋆}(HL_{2}(𝔵)) = π_{⋆}φ_{⋆}(HL_{2}(𝔵)) ⊆ π_{⋆}(HL_{2}(𝔤)).

Conversely, let 𝔮′ = Im f ⊆ 𝔮 and 𝔰 = Ker f, then the exact sequence 0 → 𝔰 → 𝔵 → 𝔮′ → 0 induces the exact sequence 0 → 𝔰/[𝔰,𝔵] = 𝔰′ → 𝔵/[𝔰,𝔵] = 𝔵′ → 𝔮′ → 0, where f′ : 𝔵′ → 𝔮′ is induced by f. Now the sequence (1) implies that f′_{⋆}(HL_{2}(𝔵′)) = f_{⋆}(HL_{2}(𝔵)) ⊆ π_{⋆}(HL_{2}(𝔤)) ⊆ HL_{2}(𝔮).

To end the proof, we need to construct φ′ : 𝔵′ → 𝔤 such that the following diagram is commutative:

[figure: 0; 𝔰′; 𝔵′; 𝔮′; 0; 0; 𝔫; 𝔤; 𝔮; 0; f′; π; φ′]

By naturality of the sequence (1) and since f′_{⋆}(HL_{2}(𝔵′)) ⊆ π_{⋆}(HL_{2}(𝔤)), there exists an injective map β : Im f′_{⋆} → Im π_{⋆} which induces τ′ : 𝔰′ → 𝔫. From Proposition 10 it follows that there exists φ′ : 𝔵′ → 𝔤; moreover φ′ is unique if and only if Hom(HL_{2}(𝔮′), 𝔫) = 0, which is obvious. Now φ : 𝔵 → 𝔤 is obtained by the composition φ′.nat : 𝔵 → 𝔵′ → 𝔤. □

Examples. Let 𝔮 be a perfect Leibniz algebra, then the universal central extension of 𝔮 is 0 → HL_{2}(𝔮) → 𝔮⋆𝔮 → 𝔮 → 0, where ⋆ denotes the non-abelian tensor product of Leibniz algebras introduced in [11]. Since a universal central extension is a stem cover (see Corollary 3 in [5]), we deduce from Corollary 1 that the map HL_{3}(𝔮 ⋆ 𝔮) → HL_{3}(𝔮) is an epimorphism and from Corollary 3 that HL_{2}(𝔮 ⋆ 𝔮) = HL_{1}(𝔮 ⋆ 𝔮) = 0.

We apply now this fact to three examples of universal central extensions:

1. Let A be an associative and unital algebra over a field K. Let stl_{n}(A) be the non commutative Steinberg algebra, which is a Leibniz algebra for n ≥ 3 [14]. Let sl_{n}(A) be the Lie algebra of matrices with entries in A whose trace in A/[A, A] is zero and let φ : stl_{n}(A) → sl_{n}(A) be the map defined by φ(v_{ij}(x)) = E_{ij}(x) where E_{ij}(x) is the matrix with only the non-zero element x in place of (i, j). For n ≥ 5 the sequence

0 → HH_{1}(A) → stl_{n}(A) →^{φ} sl_{n}(A) → 0  (4)

is the universal central extension of sl_{n}(A) in the category of Leibniz algebras [14], where HH_{1}(A) denotes the Hochschild homology group of A with coefficients in A. Applying the previous remarks to the case of the stem cover (4), we can derive the following consequences: stl_{n}(A) ≅ sl_{n}(A) ⋆ sl_{n}(A); the map HL_{3}(stl_{n}(A)) → HL_{3}(sl_{n}(A)) is an epimorphism; HL_{1}(stl_{n}(A)) = HL_{2}(stl_{n}(A)) = 0; the sequence 0 → HL_{2}(stl_{n}(A)) → HL_{2}(sl_{n}(A)) → HH_{1}(A) → 0 is exact and therefore HL_{2}(sl_{n}(A)) ≅ HH_{1}(A) (see Corollary 4.5 in [14]).

2. The universal central extension of the Lie algebra 𝔮 = Der(C[z, z^{−1}]) of derivations of Laurent polynomials in the category of Leibniz algebras is the Virasoro algebra denoted by V ir. Similarly to Example 1, we can derive now that Virasoro algebra is isomorphic to Der(C[z, z^{−1}])⋆ Der(C[z, z^{−1}]) and that the map HL_{3}(V ir) → HL_{3}(Der(C[z, z^{−1}])) is an epimorphism.

3. Let g(A) be a Kac–Moody Lie algebra. It is well-known that g(A) is a perfect Lie algebra, so it has a universal central extension in the category of Leibniz algebras. Since HL_{2}(g(A)) = 0 for any non-affine Kac-Moody Lie algebra g(A) (see [7]), it is its own universal central extension and thus 0 → 0 → g(A) → g(A) → 0 is a stem cover in the category of Leibniz algebras; hence we conclude that g(A) ≅ g(A) ⋆ g(A).

On the other hand, since HL_{2}(g(A)) ≠ 0 for any affine Kac–Moody Lie algebra g(A), the universal central extension in the category of Leibniz algebras (see [7]) is

0 → ∑_{i∈Z−{0}} kt^{ir−1} → g(A) ⊕ ∑_{i∈Z−{0}} kt^{ir−1} → g(A) → 0,

where A is of affine type X_{n}^{(r)}.

Similarly to Example 1, we can derive

g(A) ⊕ ∑_{i∈Z−{0}} kt^{ir−1} ≅ g(A) ⋆ g(A).

Moreover, the map

HL_{3} (g(A) ⊕ ∑_{i∈Z−{0}} kt^{ir−1}) → HL_{3} (g(A))

is an epimorphism.

Acknowledgements

First author was supported by Xunta de Galicia, Grant: PGIDT00PXI37101PR.

We thank the referee for several helpful suggestions which have contributed to improving the paper.

References

1. P. Batten and E. Stitzinger, On covers of Lie algebras. Comm. Algebra 24(1996), 4301–4317.

2. J. M. Casas, Central extensions of Leibniz algebras. Extracta Math. 13(1998), 393–397.

3. J. M. Casas, E. Faro, and A. M. Vieites, Abelian extensions of Leibniz algebras. Comm. Algebra 27(1999), 2833–2846.

4. J. M. Casas and T. Pirashvili, Ten–term exact sequence of Leibniz homology. J. Algebra 231 (2000), 258–264.

5. J. M. Casas and A. M. Vieites, Central extensions of perfect Leibniz algebras. Research and Expositions in Mathematics, 25 (Recent Advances in Lie Theory), 189–196, Heldermann Verlag, Berlin, 2002.

6. B. Eckmann, P. J. Hilton, and U. Stammbach, On the homology theory of central group extensions. I. The commutator map and stem extensions. Comment. Math. Helv. 47(1972), 102–122.

7. Y. Gao, The second Leibniz homology group for Kac–Moody Lie algebras. Bull. London Math. Soc. 32(2000), 25–33.

8. H. Garland, The arithmetic theory of loop groups. Inst. Hautes Études Sci. Publ. Math. No. 52, (1980), 5–136.

9. K. W. Gruenberg, Cohomological topics in group theory. Lecture Notes in Mathematics, 143. Springer–Verlag, Berlin-New York, 1970.

10. P. J. Hilton and U. Stammbach, A course in homological algebra. Graduate Texts in Mathematics, 4. Springer-Verlag, New York-Berlin, 1971.

11. R. Kurdiani and T. Pirashvili, A Leibniz algebra structure on the second tensor power. J. Lie Theory 12(2002), 583–596.

12. J.–L. Loday, Cyclic homology. Grundl. Math. Wiss. Bd. 301, Springer, 1992.

13. J.–L. Loday, Une version non commutative des algèbres de Lie: les algèbres de Leibniz. L’Enseignement Mathématique 39(1993), 269–292.

14. J.–L. Loday and T. Pirashvili, Universal enveloping algebras of Leibniz algebras and (co)homology. Math. Ann. 296(1993), 139–158.

(Received 23.05.2002)

Authors’ addresses:

J. M. Casas Dpto. Matemática Aplicada Universidad de Vigo E.U.I.T. Forestal, Campus Universitario A Xunqueira 36005 Pontevedra Spain E-mail: jmcasas@uvigo.es

M. Ladra Dpto. de Algebra Universidad de Santiago 15782 Santiago de Compostela Spain E-mail: ladra@usc.es