ABSTRACT
This chapter aims to improve, specially over fields of characteristic zero, readers understanding of the structure of the Lie algebra of an algebraic group. It also aims to use the theory of homogeneous spaces in order to complete, again in characteristic zero, the picture concerning the correspondence between properties of algebraic groups and of their associated Lie algebras. In going from Lie groups to Lie algebras, the main tool was the differentiation — linearization — process, and this technique was also used with success in the algebraic set up by the pioneers of the theory of algebraic groups. The chapter deals with the linear version of Mostow's structure theorem. Mostow's theorem asserts that over a field of characteristic zero an affine algebraic group is the semidirect product of its unipotent radical with a linearly reductive subgroup.
