## Abstract

Let g be a Leibniz algebra and E a vector space containing g as a subspace. All Leibniz algebra structures on E containing g as a subalgebra are explicitly described and classified by two non-abelian cohomological type objects: HL2g (V, g) provides the classification up to an isomorphism that stabilizes g and HL2 (V, g) will classify all such structures from the view point of the extension problem - here V is a complement

of g in E. A general product, called the unified product, is introduced as a tool for our approach. The crossed (resp. bicrossed) products between two Leibniz algebras are introduced as special cases of the unified product: the first one is responsible for the extension problem while the bicrossed product is responsible for the factorization problem. The description and the classification of all complements of a given extension g E of Leibniz algebras are given as a converse of the factorization problem. They are

classified by another cohomological object denoted by HA2(h, g |. Several examples are worked out in details.

of g in E. A general product, called the unified product, is introduced as a tool for our approach. The crossed (resp. bicrossed) products between two Leibniz algebras are introduced as special cases of the unified product: the first one is responsible for the extension problem while the bicrossed product is responsible for the factorization problem. The description and the classification of all complements of a given extension g E of Leibniz algebras are given as a converse of the factorization problem. They are

classified by another cohomological object denoted by HA2(h, g |. Several examples are worked out in details.

Original language | English |
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Pages (from-to) | 2609-2633 |

Journal | Linear Algebra and its Applications |

Volume | 439 |

Issue number | 9 |

Publication status | Published - 2013 |

## Keywords

- The extension and the factorization problem
- non-abelian cohomology for Leibniz algebras