Geometry of Central Extensions of Nilpotent Lie Algebras

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 H2(g,K)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^{2}(\mathfrak{g}, \mathbb{K})$$\end{document} of an extendable nilpotent Lie algebra g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak{g}$$\end{document} followed by studying the geometry of the orbit space of the action of the automorphism group Aut(g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak{g}$$\end{document}) on Grassmannians of the form Gr(m,H2(g,K))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\operatorname{Gr}\left(m, H^{2}(\mathfrak{g}, \mathbb{K})\right)$$\end{document}. 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.