Classifying complements for Hopf algebras and Lie algebras

Agore, A. L.; Militaru, Gigel

doi:10.1016/j.jalgebra.2013.06.012

Cited by 20 publications

(18 citation statements)

References 20 publications

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“…The present paper continues our recent work [3,4] related to the above question (3), in its general form, namely the factorization problem and its converse, called the classifying complement problem, which consist of the following question: let g ⊂ L be a given Lie subalgebra of L. If a complement of g in L exists (that is a Lie subalgebra h such that L = g + h and g ∩ h = {0}), describe explicitly, classify all complements and compute the cardinal of the isomorphism classes of all complements (which will be called the factorization index [L : g] f of g in L). Our starting point is [4,Proposition 4.4] which describes all Lie algebras L that contain a given Lie algebra h as a subalgebra of codimension 1 over an arbitrary field k: the set of all such Lie algebras L is parameterized by the space TwDer(h) of twisted derivations of h. The pioneer work on this subject was performed by K.H.…”

Section: Introductionsupporting

confidence: 86%

Bicrossed Products, Matched Pair Deformations and the Factorization Index for Lie Algebras

Agore¹,

Militaru

2014

SIGMA

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For a perfect Lie algebra h we classify all Lie algebras containing h as a subalgebra of codimension 1. The automorphism groups of such Lie algebras are fully determined as subgroups of the semidirect product h (k * × Aut Lie (h)). In the non-perfect case the classification of these Lie algebras is a difficult task. Let l(2n + 1, k) be the Lie algebra with the bracketWe explicitly describe all Lie algebras containing l(2n + 1, k) as a subalgebra of codimension 1 by computing all possible bicrossed products k l(2n + 1, k). They are parameterized by a set of matrices M n (k) 4 × k 2n+2 which are explicitly determined. Several matched pair deformations of l(2n + 1, k) are described in order to compute the factorization index of some extensions of the type k ⊂ k l(2n + 1, k). We provide an example of such extension having an infinite factorization index.

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Section: Introductionsupporting

confidence: 86%

Bicrossed Products, Matched Pair Deformations and the Factorization Index for Lie Algebras

Agore¹,

Militaru

2014

SIGMA

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“…Recently [5] the following problem called the extending structures problem, was addressed at the level of Leibniz algebras generalizing the one from Lie algebras [4] (see also [1,2,3]). Equivalently restated, it consists of the following question: let g be a Leibniz algebra, E a vector space and i : g → E a linear injective map.…”

Section: Introductionmentioning

confidence: 99%

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

Militaru

2014

Linear and Multilinear Algebra

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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

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“…The classifying complements problem (CCP) was introduced in [2] in a very general, categorical setting, as a sort of converse of the factorization problem. A similar problem, called invariance under twisting, was studied in [11] for Brzezinski's crossed products.…”

Section: Introductionmentioning

confidence: 99%

Classifying complements for associative algebras

Agore

2014

Linear Algebra and its Applications

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Abstract. For a given extension A ⊂ E of associative algebras we describe and classify up to an isomorphism all A-complements of E, i.e. all subalgebras X of E such that E = A+X and A∩X = {0}. Let X be a given complement and (A, X, ⊲, ⊳, ↼, ⇀ the canonical matched pair associated with the factorization E = A + X. We introduce a new type of deformation of the algebra X by means of the given matched pair and prove that all A-complements of E are isomorphic to such a deformation of X. Several explicit examples involving the matrix algebra are provided.

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Classifying complements for Hopf algebras and Lie algebras

Cited by 20 publications

References 20 publications

Bicrossed Products, Matched Pair Deformations and the Factorization Index for Lie Algebras

Bicrossed Products, Matched Pair Deformations and the Factorization Index for Lie Algebras

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

Classifying complements for associative algebras

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