Geometry of Central Extensions of Nilpotent Lie Algebras

Abstract

We obtain a recurrent and monotone method for constructing and classifying nilpotent Lie algebras by means of successive central extensions. The method consists in calculating the second cohomology \(H^{2}(\mathfrak{g}, \mathbb{K})\) of an extendable nilpotent Lie algebra \(\mathfrak{g}\) followed by studying the geometry of the orbit space of the action of the automorphism group Aut(\(\mathfrak{g}\)) on Grassmannians of the form \(\operatorname{Gr}\left(m, H^{2}(\mathfrak{g}, \mathbb{K})\right)\). In this case, it is necessary to take into account the filtered cohomology structure with respect to the ideals of the lower central series: a cocycle defining a central extension must have maximum filtration. Such a geometric method allows us to classify nilpotent Lie algebras of small dimensions, as well as to classify narrow naturally graded Lie algebras. We introduce the concept of a rigid central extension and construct examples of rigid and nonrigid central extensions.