Nilpotent and solvable lie algebras

Abstract

This chapter discusses finite-dimensional solvable and nilpotent Lie algebras. These classes of Lie algebras play an important role in the structure theory. A Lie algebra over a field of characteristic 0 is presented as a semi-direct sum of a semi-simple subalgebra and its radical. The classical results of that give a complete classification of semi-simple Lie algebras are focused. The structure theory of Lie algebras is reduced to the study of solvable Lie algebras. The class of nilpotent Lie algebras is a very important subclass in the class of the solvable ones. The problem of their description is reduced to a description of the nilpotent ones. The first non-trivial classifications of some classes of low-dimensional nilpotent Lie algebras are because of Umlauf. The list of nilpotent Lie algebras of dimension m ≤ 6 is presented. Umlauf's list of filiform Lie algebras is exact only for dimension m ≤ 7; in dimension 8 and 9 the list contains errors and it is incomplete. The list of nilpotent Lie algebras of dimension ≤ 6 also contains errors.