The global extension problem, co-flag and metabelian Leibniz algebras

Abstract

Let $\mathfrak{L}$ be a Leibniz algebra, $E$ a vector space and $\pi : E \to \mathfrak{L}$ an epimorphism of vector spaces with $ \mathfrak{g} = {\rm Ker} (\pi)$. The global extension problem asks for the classification of all Leibniz algebra structures that can be defined on $E$ such that $\pi : E \to \mathfrak{L}$ is a morphism of Leibniz algebras: from a geometrical viewpoint this means to give the decomposition of the groupoid of all such structures in its connected components and to indicate a point in each component. All such Leibniz algebra structures on $E$ are classified by a global cohomological object ${\mathbb G} {\mathbb H} {\mathbb L}^{2} \, (\mathfrak{L}, \, \mathfrak{g})$ which is explicitly constructed. It is shown that ${\mathbb G} {\mathbb H} {\mathbb L}^{2} \, (\mathfrak{L}, \, \mathfrak{g})$ is the coproduct of all local cohomological objects $ {\mathbb H} {\mathbb L}^{2} \, \, (\mathfrak{L}, \, (\mathfrak{g}, [-,-]_{\mathfrak{g}}))$ that are classifying sets for all extensions of $\mathfrak{L}$ by all Leibniz algebra structures $(\mathfrak{g}, [-,-]_{\mathfrak{g}})$ on $\mathfrak{g}$. The second cohomology group ${\rm HL}^2 \, (\mathfrak{L}, \, \mathfrak{g})$ of Loday and Pirashvili appears as the most elementary piece among all components of ${\mathbb G} {\mathbb H} {\mathbb L}^{2} \, (\mathfrak{L}, \, \mathfrak{g})$. Several examples are worked out in details for co-flag Leibniz algebras over $\mathfrak{L}$, i.e. Leibniz algebras $\mathfrak{h}$ that have a finite chain of epimorphisms of Leibniz algebras $\mathfrak{L}_n : = \mathfrak{h} \stackrel{\pi_{n}}{\longrightarrow} \mathfrak{L}_{n-1} \, \cdots \, \mathfrak{L}_1 \stackrel{\pi_{1}} {\longrightarrow} \mathfrak{L}_{0} := \mathfrak{L}$ such that ${\rm dim} ({\rm Ker} (\pi_{i})) = 1$, for all $i = 1, \cdots, n$.Comment: 20 pages: Continues arXiv:1307.254

On a type of commutative algebras

“…As we shall see, CF (A) parameterizes the set of all crossed systems of A by a 1-dimensional vector space. Our next result provides a description of the first algebra A 1 in the exact sequence (20) in terms depending only on A.…”

“…In Section 2 we deal with the main question addressed in this paper, namely the global extension (GE) problem, and the crossed product for JJ algebras is introduced as the object responsible for it. Introduced for Leibniz algebras in [20] and studied for associative/Poisson algebras in [1,2], the GE problem is a generalization of the classical Hölder's extension problem, and for JJ algebras consists of the following question:…”

Hochschild products and global non-abelian cohomology for algebras. Applications

“…Hence, in order for ϕ to be an algebra map it should take the following simplified form for any a ∈ A and x ∈ k: ϕ(a, x) = ψ(a), r(a) + x s 0 (33) for some triple (s 0 , ψ, r) ∈ k × Hom k (A, A) × Hom k (A, k). Next we prove that a linear map given by (33) is an algebra morphism from A (λ,Λ,ϑ) to A (λ ′ ,Λ ′ ,ϑ ′ ) if and only if ψ : A → A is an algebra map and the following compatibilities are fulfilled for any a, b ∈ A:…”

Itô’s theorem and metabelian Leibniz algebras

“…We denote by P (λ, θ) the associated metabelian product k#P . Using, (27)- (28) we can easily see that there exists an isomorphism of Leibniz algebras P (λ, θ) ∼ = P (λ, 0) , for all λ ∈ P * \ {0} and θ ∈ P * (take u := 1, ψ := Id P and v := θ in (27)- (28)). Thus, we have obtained that for Λ = 0 and λ = 0, the corresponding metabelian product P (λ, Λ, f ) is isomorphic to the Leibniz algebra P λ from (2).…”