# Freudenthal triple systems via root system methods Open Access

## Helenius, Fred William (2009)

Permanent URL: https://etd.library.emory.edu/concern/etds/05741s219?locale=en
Published

## Abstract

For a Lie algebra 𝔤 of type B, D, E or F, we can apply a grading 𝔤 = 𝔤−2⊕𝔤−1⊕𝔤0⊕𝔤1⊕𝔤2 and then define a quartic form and a skew-symmetric bilinear form on 𝔤1, 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 important cases 𝔤= E8 and 𝔤 = D4, we determine the groups stabilizing the quartic form and both the quartic and bilinear forms.

1 Introduction 1
1.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Prior work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Technical background 8
2.1 Root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Classification of root systems . . . . . . . . . . . . . . . . . . 12
2.3 Structure constants . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4 Our situation . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.5 Roots of α-height 1 . . . . . . . . . . . . . . . . . . . . . . . . 21
2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3 General Results 27
3.1 The bilinear and quartic forms . . . . . . . . . . . . . . . . . . 27
3.2 Strictly regular elements . . . . . . . . . . . . . . . . . . . . . 34
3.3 Freudenthal triple systems . . . . . . . . . . . . . . . . . . . . 44
3.4 Computation of the 4-linear form . . . . . . . . . . . . . . . . 45
4 Special Results 51
4.1 Eigenspace decomposition of 𝔤1 . . . . . . . . . . . . . . . . . 51
4.2 Characterization of the orbits . . . . . . . . . . . . . . . . . . 53

4.3 Related groups . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.4 The stabilizer of the quartic form: G = E8 . . . . . . . . . . . 56
4.5 The stabilizer of the quartic form: G = D4 . . . . . . . . . . . 66
5 Conclusion 73
5.1 Summary of results . . . . . . . . . . . . . . . . . . . . . . . . 73
5.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Bibliography 77