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

Imaginary Killing Spinors in Lorenztian Geometry

We study the geometric structure of Lorentzian spin manifolds, which admit imaginary Killing spinors. The discussion is based on the cone construction and a normal form classification of skew-adjoint operators in signature $(2,n-2)$. Derived geometries include Brinkmann spaces, Lorentzian Einstein-Sasaki spaces and certain warped product structures. Exceptional cases with decomposable holonomy of the cone are possible.

Fixed points subgroups $G^{σ,σ'}$ by two involutive automorphisms $σ$, $σ'$ of exceptional compact Lie group $G$, Part II, $G = E_8$

For the simply connected compact exceptional Lie group $E_8$, we determine the structure of subgroup $(E_8)^{σ, σ'}$ of $E_8$ which is the intersection $(E_8)^σ\cap (E_8)^{σ'}$. Then the space $E_8/(E_8)^{σ, σ'}$ is the exceptional $\mathbb{Z}_2 \times \mathbb{Z}_2$- symmetric space of type EVIII-VIII-VIII, and that we give two involutions $σ, σ'$ for the space $E_8/(E_8)^{σ, σ'}$ concretely.

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}…

Spinor-generators of compact exceptional Lie groups

We know that any element A of the group SO(3) can be represented as A = A1 A2 A1', where A1, A1' are elements of SO1(2)={A is an element of SO(3) | Ae1=e1}, and SO2(2)={A is an element of SO(3) | Ae2=e2} . This fact is known as Euler's angle. When this situation, a matrix A is called the generator. In the present paper, we shall show firstly that the similar results hold for the groups SU(3), and Sp(3). Secondly, we shall show that any element g of the simply connected compact Lie group F4 (respectively. E6) can be represented g= g1 g2 g1', where g1, g1' are elements of Spin1(9), g2 is an elem…

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…

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。