Some properties of the space of $n$-dimensional Lie algebras

Some general properties of the space $\mathscr L_n$ of $n$-dimensional Lie algebras are studied. This space is defined by the system of Jacobi's quadratic equations. It is proved that these equations are linearly independent and equivalent to each other (more precisely, the quadratic forms defining these equations are affinely equivalent). Moreover, the problem on the closures of some orbits of the natural action of the group $\mathrm{GL}_n$ on $\mathscr L_n$ is considered. Two Lie algebras are indicated whose orbits are closed in the projectivization of the space $\mathscr L_n$. The intersection of all irreducible components of the space $\mathscr L_n$ is also treated. It is proved that this intersection is nontrivial and consists of nilpotent Lie algebras. Two Lie algebras belonging to this intersection are indicated. Some other results concerning arbitrary Lie algebras and the space $\mathscr L_n$ formed by these algebras are presented.
Bibliography: 17 titles.