Affine structures on three-step nilpotent Lie algebras

Source abstract

As part of the investigation of the parallel between solvable Lie theory and the theory of groups of affine motions, it is proved that every three-step nilpotent Lie algebra admits a faithful representation of a certain special kind. It follows immediately that every three-step nilpotent Lie group which is connected, simply connected, and of dimension n n admits a representation as a simply transitive group of affine motions of R n {R^n} .