Extending structures for Lie algebras

Abstract

Abstract. Let g be a Lie algebra, E a vector space containing g as a subspace. The paper is devoted to the extending structures problem which asks for the classification of all Lie algebra structures on E such that g is a Lie subalgebra of E. A general product, called the unified product, is introduced as a tool for our approach. Let V be a complement of g in E: the unified product g ♮ V is associated to a system (⊳, ⊲, f, {−, −}) consisting of two actions ⊳ and ⊲, a generalized cocycle f and a twisted Jacobi bracket {−, −} on V . There exists a Lie algebra structure [−, −] on E containing g as a Lie subalgebra if and only if there exists an isomorphism of Lie algebras (E, [−, −]) ∼ = g ♮ V . All such Lie algebra structures on E are classified by two cohomological type objects which are explicitly constructed. The first one H 2 g (V, g) will classify all Lie algebra structures on E up to an isomorphism that stabilizes g while the second object H 2 (V, g) provides the classification from the view point of the extension problem. Several examples that compute both classifying objects H 2 g (V, g) and H 2 (V, g) are worked out in detail in the case of flag extending structures.

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

“…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. Hofmann: [12, Theorem I] describes the structure of n-dimensional real Lie algebras containing a given subalgebra of dimension n − 1.…”

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

“…The formula for the comultiplication of any such coalgebra (C * ⋆ k n ) * can be written down effectively by using (41) and (3). This observation shows that the GE-problem applied for finite dimensional algebras and finite dimensional vector spaces gives the answer at the level of finite dimensional coalgebras to what we have called the extending structures problem studied in [2,4,5] for Jacobi, Lie and respectively associative algebras.…”

Jacobi and Poisson algebras

“…Indeed, we will fix {1, x} as a basis in a two dimensional Jacobi algebra. The proof follows from the classical classification of 2-dimensional Lie algebras [34] and from the well known classification of 2-dimensional associative algebras [46] (for arbitrary fields see [7,Corollary 4.5]). Indeed, the classification follows by a routine computation based on checking the compatibility condition (6).…”