Proceedings of the Steklov Institute of Mathematics | 2019
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 $$H^{2}(\\mathfrak{g}, \\mathbb{K})$$H2(g,K) of an extendable nilpotent Lie algebra $$\\mathfrak{g}$$g followed by studying the geometry of the orbit space of the action of the automorphism group Aut($$\\mathfrak{g}$$g) on Grassmannians of the form $$\\operatorname{Gr}\\left(m, H^{2}(\\mathfrak{g}, \\mathbb{K})\\right)$$Gr(m,H2(g,K)). 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.