ABSTRACT
Leibniz algebras were introduced by Bloh [51] under the name of D-algebras and rediscovered later on by Loday [166] as non-commutative generalizations of Lie algebras. This chapter studies the extending structures (ES) problem for Leibniz algebras; more precisely, for a given Leibniz algebra g and a vector space E containing g as a subspace, the set of all Leibniz algebra structures that can be defined on E such that g becomes a Leibniz subalgebra of E are described and classified. The construction responsible for providing the answer to the ES problem is called unified product of Leibniz algebras.
Several special cases are discussed and explicit examples are given, including the bicrossed product of Leibniz algebras which is the main tool for addressing the factorization problem.
The theoretical answer to the classification part of the ES problem is given by explicitly constructing a a cohomological type group; the classification is given up to an isomorphism of Leibniz algebras which stabilizes g. A second classifying object is also constructed: it provides the classification from the point of view of the extension problem.
The chapter ends with a discussion of Ito's theorem in the context of Leibniz algebras.
