Freudenthal triple systems by root system methods

TL;DR

This paper explores the construction and analysis of Freudenthal triple systems within certain Lie algebras using root system techniques, focusing on specific cases like E_8 and D_4 to identify stabilizing groups.

Contribution

It introduces a root system approach to study Freudenthal triple systems derived from Lie algebra gradings, providing new insights into their structure and symmetry groups.

Findings

Determined stabilizer groups for E_8 and D_4 cases.

Established a method to construct Freudenthal systems via root systems.

Analyzed the structure of these systems using Lie algebra techniques.

Abstract

For certain Lie algebras g, we can use a Z/5Z-grading and define a quartic form and a skew-symmetric bilinear form on the degree 1 component, g_1, thereby constructing a Freudenthal triple system. The structure of the Freudenthal triple system is examined using root system methods available in the Lie algebra context. In the cases g = E_8 (where g_1 is the minuscule representation of E_7) and g = D_4, we determine the groups stabilizing the quartic form and both the quartic and bilinear forms.