A geometric construction of the exceptional Lie algebras F4 and E8

Adjoint orbit types of compact exceptional Lie group G2 in its Lie algebra

A Lie group $G$ naturally acts on its Lie algebra $\gg$, called the adjoint action. In this paper, we determine the orbit types of the compact exceptional Lie group $G_2$ in its Lie algebra $\gg_2$. As results, the group $G_2$ has four orbit types in the Lie algebra $\gg_2$ as $$ G_2/G_2, \quad G_2/(U(1) \times U(1)), \quad G_2/((Sp(1)\times U(1))/\Z_2), \quad G_2/((U(1)\times Sp(1))/\Z_2). $$ These orbits, especially the last two orbits, are not equivalent, that is, there exists no $G_2$-equivariant homeomorphism among them.

Low Cohomogeneity and Polar Actions on Exceptional Compact Lie Groups

We study isometric Lie group actions on the compact exceptional groups E6, E7, E8, F4 and G2 endowed with a biinvariant metric. We classify polar actions on these groups. We determine all isometric actions of cohomogeneity less than three on E6, E7, F4 and all isometric actions of cohomogeneity less than 20 on E8. Moreover we determine the principal isotropy algebras for all isometric actions on G2.

Decomposition of compact exceptional Lie groups into their maximal tori

In this paper we treat the intersection of fixed point subgroups by the involutive automorphisms of exceptional Lie group $G= F_4, E_6, E_7$. We shall find involutive automorphisms of $G$ such that the connected component of the intersection of those fixed point subgroups coincides with the maximal torus of $G$.

Octonions, triality, the exceptional Lie algebra F4, and polar actions on the Cayley hyperbolic plane

Using octonions and the triality property of Spin(8), we find explicit formulae for the Lie brackets of the exceptional simple real Lie algebras $\mathfrak{f}_4$ and $\mathfrak{f}^*_4$, i.e. the Lie algebras of the isometry groups of the Cayley projective plane and the Cayley hyperbolic plane. As an application, we classify polar actions on the Cayley hyperbolic plane which leave a totally geodesic subspace invariant.

New Non-naturally reductive Einstein metrics on Exceptional simple Lie groups

In this article, we achieved several non-naturally reductive Einstein metrics on exceptional simple Lie groups, which are formed by the decomposition arising from general Wallach spaces. By using the decomposition corresponding to the two involutive automorphisms, we calculated the non-zero coefficients in the expression for the components of Ricci tensor with respect to the given metrics. The Einstein metrics are obtained as solutions of systems polynomial equations, which we manipulate by symbolic computations using Gröbner bases.

Maximal representations of uniform complex hyperbolic lattices in exceptional Hermitian Lie groups

We complete the classification of maximal representations of uniform complex hyperbolic lattices in Hermitian Lie groups by dealing with the exceptional groups ${\rm E}_6$ and ${\rm E}_7$. We prove that if $ρ$ is a maximal representation of a uniform complex hyperbolic lattice $Γ\subset{\rm SU}(1,n)$, $n>1$, in an exceptional Hermitian group $G$, then $n=2$ and $G={\rm E}_6$, and we describe completely the representation $ρ$. The case of classical Hermitian target groups was treated by Vincent Koziarz and the second named author (arxiv:1506.07274). However we do not focus immediately on the ex…

Realizations of globally exceptional $\mathbb{Z}_2 \times \mathbb{Z}_2$- symmetric spaces

A classification is given of the exceptional $\mathbb{Z}_2 \times \mathbb{Z}_2$-symmetric spaces $G/K$ by A.Kollross, where $G$ is an exceptional compact Lie group or $S\!pin(8)$, and moreover the structure of $K$ is determined as Lie algebra. In the present article, we give a pair of commuting involutive automorphisms (involutions) $\tildeσ, \tildeτ$ of $G$ concretely and determine the structure of group $G^σ \cap G^τ$ corresponding to Lie algebra $\mathfrak{g}^σ\cap \mathfrak{g}^τ$, where $G$ is an exceptional compact Lie group. Thereby, we realize exceptional $\mathbb{Z}_2 \times \mathbb{Z}…

Non-existence of standard compact Clifford-Klein forms of homogeneous spaces of exceptional Lie groups

We use a computer-aided approach to prove that there are no standard compact Clifford-Klein forms of homogeneous spaces of exceptional Lie groups. This yields further support for Kobayashi's conjecture about possible compact Clifford-Klein forms. On one hand, our approach is based on the algorithms developed in this work which eliminate the majority of possibilities. On the other hand, we complete the proof using the algorithmic methods of classifying semisimple subalgebras in simple real Lie algebras developed by Faccin and de Graaf, as well as by calculating invariants like a-hyperbolic rank…

Non-naturally reductive Einstein metrics on exceptional Lie groups

Given an exceptional compact simple Lie group $G$ we describe new left-invariant Einstein metrics which are not naturally reductive. In particular, we consider fibrations of $G$ over flag manifolds with a certain kind of isotropy representation and we construct the Einstein equation with respect to the induced left-invariant metrics. Then we apply a technique based on Gröbner bases and classify the real solutions of the associated algebraic systems. For the Lie group ${\rm G}_2$ we obtain the first known example of a left-invariant Einstein metric, which is not naturally reductive. Moreover, f…

Decomposition of spinor groups by the involution σ' in exceptional Lie groups

The compact exceptional Lie groups F4, E6, E7 and E8 have spinor groups as a subgroup as follows: E8 \supset Ss(16) \supset Spin(15) \supset Spin(14) \supset Spin(13), E7 \supset Spin(12) \supset Spin(11), E6 \supset Spin(10), F4 \supset Spin(9) \supset Spin(8) \supset Spin(7) \supset \cdot \cdot \cdot \supset Spin(1) \ni 1. We know the involution σ' induced an element σ' \in Spin(8) \subset F4 \subset E6 \subset E7 \subset E8. Now, in this paper, we determine the group structures of (Spin(n))σ' which are the fixed subgroups by the involution σ'.

与例外李超代数相关的特殊正交表示的几何性质

Geometric properties of special orthogonal representations associated to exceptional Lie superalgebras

本文基于特征不为2或3的域上的八元数代数，证明了导子代数与特殊正交李代数的基础表示和旋量表示为特殊正交表示，揭示了其与二元三次式相关的几何性质及与Fano平面的联系，并给出了例外李超代数的构造。

E8 例外李代数的括号

The bracket of the exceptional Lie algebra E8

基于三重性与八八元数，研究E8李代数的括号公式，并推导E6、E7子代数及F4的26维不可约表示公式。

Fixed points subgroups by two involutive automorphisms $σ, γ$ of compact exceptional Lie groups $F_4, E_6$ and $E_7$

For simply connected compact exceptional Lie groups $G = F_4, E_6$ and $E_7$, we consider two involutions $σ, γ$ and determine the group structure of subgroups $G^{σ,γ}$ of $G$ which are the intersection $G^σ\cap G^γ$ of the fixed points subgroups of $G^σ$ and $G^γ$. The motivation is as follows. In [1](see the References of this paper), we determine the group structure of $(F_4)^{σ, σ'}, (E_6)^{σ, σ'}$ and $(E_7)^{σ, σ'}$, and in [2](see the References of this paper), we also determine the group structure of $(G_2)^{γ, γ'}, (F_4)^{γ, γ'}$ and $(E_6)^{γ, γ'}$. So, in this paper, we try to dete…

Hilbert空间上极作用不存在例外轨道

Non-existence of exceptional orbits under polar actions on Hilbert spaces

本文证明可分Hilbert空间上连通Hilbert Lie群的极作用无例外轨道，推广有限维结果，并给出紧致对称空间超极作用无例外轨道的简单几何证明。

Realizations of inner automorphisms of order four and fixed points subgroups by them on the connected compact exceptional Lie group $E_8$, Part II

The compact simply connected Riemannian 4-symmetric spaces were classified by J.A. Jim{é}nez according to type of the Lie algebras. As homogeneous manifolds, these spaces are of the form $G/H$, where $G$ is a connected compact simple Lie group with an automorphism $\tildeγ$ of order four on $G$ and $H$ is a fixed points subgroup $G^γ$ of $G$. According to the classification by J.A. Jim{é}nez, there exist seven compact simply connected Riemannian 4-symmetric spaces $ G/H $ in the case where $ G $ is of type $ E_8 $. In the present article, %as Part II continuing from Part I, for the connected c…

$T$-root systems of exceptional flag manifolds. Classification and invariants

We classify the systems of $T$-roots of the flag manifolds $M$ of the exceptional compact simple Lie groups with the second Betti number $b_2(M)\ge2$.

Orbit types of the compact Lie group E_7 in the complex Freudenthal vector space P^C

Let J be the exceptional Jordan algebra over R and J^C its complexification. Then the simply connected compact exceptional Lie group F_4 acts on J and F_4 has three orbit types which are F_4/F_4, F_4/Spin(9), F_4/Spin(8). Similarly the simply connected compact exceptional Lie group E_6 acts on J^C and E_6 has five orbit types which are E_6/E_6, E_6/F_4, E_6/Spin(10), E_6/Spin(9), E_6/Spin(8) . In this paper, we determine the orbit types of the simply connected compact exceptional Lie group E_7 in the complex Freudenthal vector space P^C. As results, E_7 has seven orbit types which are E_7/E_7,…

Exceptional (Z/2Z) x (Z/2Z)-symmetric spaces

The notion of (Z/2Z) x (Z/2Z)-symmetric spaces is a generalization of classical symmetric spaces, where the group Z/2Z is replaced by (Z/2Z) x (Z/2Z). In this article, a classification is given of the (Z/2Z) x (Z/2Z)-symmetric spaces G/K where G is an exceptional compact Lie group or Spin(8), complementing recent results of Bahturin and Goze. Our results are equivalent to a classification of (Z/2Z) x (Z/2Z)-gradings on the exceptional simple Lie algebras e6, e7, e8, f4, g2 and so(8).

Realization of globally exceptional Riemannian $4$-symmetric space $E_8/(E_8)^{σ'_{4}}$

The compact simply connected Riemannian 4-symmetric spaces were classified by J.A. Jim{é}nez. As homogeneous manifolds, these spaces are of the $G/H$, where $G$ is a connected compact simple Lie group with an automorphism $\tildeγ$ of oder 4 and $H$ is a fixed points subgroup $G^γ$ of $G$. In the present article, for the exceptional compact Lie group $G=E_8$, we give the explicit form of automorphism of order 4 induced by the $\boldsymbol{R}$-linear transformation $σ'_{4}$ and determine the structure of the group $(E_8)^{σ'_{4}}$. Thereby, we realize the globally exceptional Riemannian $4$-sym…