On the Amount of Nondegenerate Tubular Orbits of 7-Dimensional Lie Algebras in \(\mathbb C^4\)

Abstract

We consider holomorphic realizations in \(\mathbb C^4\) for a large family of 7-dimensional Lie algebras containing a 6-dimensional nilradical and one or two 4-dimensional abelian subalgebras. We show that for these Lie algebras, a natural condition of having tubular Levi-nondegenerate 7-dimensional orbits is rarely compatible with a straightened basis of one of the abelian subalgebras. In many cases, this incompatibility follows easily from the structure and properties of abelian ideals in 4-dimensional subalgebras of the algebras in question.