Title: Graded Lie algebras and regular prehomogeneous vector spaces with one-dimensional scalar multiplication,
Abstract: The aim of this paper is to study relations between regular reductive PVs
with one-dimensional scalar multiplication and the structure of graded Lie
algebras. We will show that the regularity of such PVs is described by an
$\mathfrak{sl}_2$-triplet of a graded Lie algebra.