Affine structures on 4-step nilpotent lie algebras Original Research Article

In this paper we construct complete left-symmetric (and so affine) structures on a class of 4-step nilpotent Lie algebras. To achieve this, we use the language of derivations and translate the problem of the existence of a complete left-symmetric structure for a given Lie algebra L, to the existence of a certain submodule (called layerwise complementary) of the augmentation ideal of the universal enveloping algebra of L. A polynomial construction (an analogue of the classical polynomial construction for groups), due to the second author, is used to determine such a submodule for all 3-step nilpotent Lie algebras (allowing to rediscover the known results) and for a reasonable class of 4-step nilpotent Lie algebras (for which the existence of a complete left-symmetric/affine structure was not known before). A concrete description of this left-symmetric structure is given.