Klein four-subgroups of Lie algebra automorphisms

Abstract

By calculating the symmetric subgroups Aut(u 0 ) θ and their involution classes, we classify the Klein four subgroups Γ of Aut(u 0 ) for each compact simple Lie algebra u 0 up to conjugation. This leads to a new approach of classification of semisimple symmetric pairs and Z 2 × Z 2 -symmetric spaces. We also determine the fixed point subgroup Aut(u 0 ) Γ .

Classification of Klein four symmetric pairs of holomorphic type for $$\mathrm {E}_{6(-14)}$$ E 6 ( - 14 )

“…Jingsong HUANG and Jun YU studied semisimple symmetric spaces from a different point of view in [7]; that is, by determining the Klein four subgroups in the automorphism groups of compact Lie algebras. As an extension of the work of [7], Jun YU classified all the elementary abelian 2-groups in the automorphism groups of compact Lie algebras, where groups of rank 2 are just Klein four subgroups. However, the classification of Klein four subgroups in the automorphism groups of non-compact simple Lie algebras is not clear.…”

“…Let G = Autu 0 , and G 0 = Intu 0 be the identity component of G. From [7], it is known that (G 0 ) σ3 ∼ = F 4(−52) , the compact Lie group of type F 4 , and there exist involutive automorphisms τ 1 and τ 2 of F 4(−52) such that f By [7], τ 1 , τ 2 , σ 3 τ 1 , and σ 3 τ 2 represent all conjugacy classes of involutive automorphisms in G σ3 except for σ 3 and there are conjugacy relations in G:…”

“…Let η 1 = iI, η 2 = −I 2 0 0 I 2 , and η 3 = −1 0 0 I 3 . By [7], η 1 , η 2 , η 3 , σ 4 η 1 , σ 4 η 2 , and σ 4 η 3 represent all conjugacy classes of involutive automorphisms in G σ4 except for σ 3 and there are conjugacy relations in G: 3.2. Elements conjugate to σ 2 .…”

“…Firstly, x 1 ∼ G σ 1 by [15]. Secondly, x 2 is conjugate to τ 1 in F 4(−52) by [7], and it follows that x 2 ∼ G σ 1 . Thirdly, since it is well known that ±i, ±j, and ±k are pairwisely conjugate in the group ring of the quoternion group, it is immediate that…”

“…Firstly, x 4 is conjugate to τ 2 in F 4(−52) by [7], and it follows that x 4 ∼ G σ 2 . Secondly, it is obvious that x 4 ∼ H x 5 ∼ H x 4 x 5 .…”