3-graded decompositions of exceptional Lie algebras g and group realizations of gev, g0 and ged, III: G=E8

Toshikazu Miyashita,Ichiro Yokota

doi:10.1215/0023608x-2009-014

Abstract

In the articles [4] and [7], we completed the determination of group realizations gev and g0 of 2-graded decompositions g=g-2⊕g-1⊕g0⊕g1⊕g2 of exceptional Lie algebras g for the universal exceptional Lie groups. In the present article, which is a continuation of [5] and [8], we determine group realizations of subalgebras gev, g0 and ged of 3-graded decompositions of exceptional Lie algebras g for the universal exceptional Lie groups of type E8.

Highlights

The 3-graded decompositions of simple Lie algebras g, g = g−3 ⊕ g−2 ⊕ g−1 ⊕ g0 ⊕ g1 ⊕ g2 ⊕ g3, [gk, gl] ⊂ gk+l, are classified, and the types of subalgebras gev = g−2 ⊕ g0 ⊕ g2, g0 and ged = g−3 ⊕ g0 ⊕ g3 are determined
For the connected exceptional universal linear Lie groups G of type E8, we realize the subgroups Gev, G0, and Ged of G corresponding to gev, g0, and ged of g = Lie G
As we use a realization of semispinor groups Ss(16, C) in E8C and Ss(8, 8) in E8(8) by Gomyo [2], we review here one more Lie algebra e8C constructed by Gomyo [2]

Summary

Introduction

The connected universal linear Lie groups E8C , E8(8), and E8(−24) of type E8 are given, respectively, by (2) Since the group E7C has subgroups C∗ and E6C (see [6, Theorem 4.4.4]), we define a mapping φ0 : SL(2, C) × C∗ × E6C → (E8C )υι = (E8C )0 by φ0(A, θ, β) = ψ(A)φ(θ)β as the restriction mapping of φev.

Results

Conclusion