AFFINE STRUCTURES ON 4-STEP NILPOTENT LIE-ALGEBRAS

In this paper we construct complete left-symmetric (and so affine) str uctures on a class of 4-step nilpotent Lie algebras. To achieve this, we use the language of derivations and translate the problem of the ex istence of a complete left-symmetric structure for a given Lie algebra L, to the existence of a certain submodule (called layerwise compleme ntary) of the augmentation ideal of the universal enveloping algebra o f L. A polynomial construction (an analogue of the classical polynomia l construction for groups), due to the second author, is used to deter mine 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-sym metric/affine structure was not known before). A concrete description of this left-symmetric structure is given. (C) 1997 Elsevier Science B .V.