# Graded Lie algebras and regular prehomogeneous vector spaces with one-dimensional scalar multiplication

## TL;DR

This paper explores the connection between regular prehomogeneous vector spaces with one-dimensional scalar multiplication and graded Lie algebras, revealing that their regularity is characterized by an sl_2-triplet within the graded Lie algebra structure.

## Contribution

It establishes a link between the regularity of certain prehomogeneous vector spaces and the structure of graded Lie algebras via sl_2-triplets, providing new insights into their classification.

## Key findings

- Regular PVs with one-dimensional scalar multiplication are characterized by sl_2-triplets.
- The regularity condition is described through the structure of graded Lie algebras.
- The study offers a structural criterion for the regularity of these PVs.

## 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.