ABSTRACT
This chapter is mainly concerned with the extending structures (ES) problem for Lie algebras. The unified product for Lie algebras is introduced as the construction used to deal with the ES problem. Several results concerning the ES problem are derived as special cases of their counterparts proved for Leibniz algebras in the previous chapter. A method of computing the classification object for the ES problem in the case of the so called flag extending structures is explicilty described.
Some special cases of the unified product are discussed as well, such as: the crossed product, the bicrossed product and the skew crossed product which plays the key role in developing a Galois theory for Lie algebras. The next two sections of this chapter contain some explicit computations concerning the factorization problem and its converse, the classifying complements problem.
The last part of this chapter develops a Galois theory for Lie algebras. Aside from describing the Galois group of a Lie algebra extension as a subgroup of a certain semidirect product of groups, the counterparts of Artin's Theorem for Lie algebras and Hilbert's 90 Theorem are proved.
