例外紧李群PDMATHAAD的两个对合自同构PDMATHAAB，PDMATHAAC的不动点子群$G^{σ,σ'}$，Part II，$G = E_8$

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}…

Symmetric spaces with dissecting involutions

An involutive diffeomorphism $σ$ of a connected smooth manifold $M$ is called dissecting if the complement of its fixed point set is not connected. Dissecting involutions on a complete Riemannian manifold are closely related to constructive quantum field theory through the work of Dimock and Jaffe/Ritter on the construction of reflection positive Hilbert spaces. In this article we classify all pairs $(M,σ)$, where $M$ is an irreducible symmetric space, not necessarily Riemannian, and $σ$ is a dissecting involutive automorphism. In particular, we show that the only irreducible $1$-connected Rie…

例外李群

Exceptional Lie groups

我们用非常初等的方法描述了单连通紧例外单李群。首先具体构造了所有单连通紧例外李群G.其次，我们找到了G的所有对合自同构，并确定了其不动点子群的群结构。它们对应于例外型的所有不可约紧对称空间的分类，也对应于所有非紧例外单李群的分类。最后，我们确定了极大秩的极大子群的群结构。无论如何，我们希望这本书能被用于数学和物理学。

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…

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

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…

Integrability of geodesic flows for metrics on suborbits of the adjoint orbits of compact groups

Let $G/K$ be an orbit of the adjoint representation of a compact connected Lie group $G$, $σ$ be an involutive automorphism of $G$ and $\tilde G$ be the Lie group of fixed points of $σ$. We find a sufficient condition for the complete integrability of the geodesic flow of the Riemannian metric on $\tilde G/(\tilde G\cap K)$, which is induced by the bi-invariant Riemannian metric on $\tilde G$. The integrals constructed here are real analytic functions, polynomial in momenta. It is checked that this sufficient condition holds when $G$ is the unitary group $U(n)$ and $σ$ is its automorphism defi…

Banach对称空间

Banach Symmetric Spaces

在O. Loos是一个光滑的巴拿赫流形$M$，它具有一个乘法映射$μ\colon M \times M \to M$，使得每个左乘法映射$μ_x := μ(x,\cdot)$（与$x \in M$）是$(M,μ)$与孤立不动点$x$的对合自同构.我们证明了对称空间的李三系的态射可以被唯一地积分，只要第一流形是1-连通的。证明了在挠率和曲率平行的仿射巴拿赫流形的切空间之间的一个连续线性映射是一个仿射映射的可积性，如果它使挠率张量和曲率张量相互缠绕，则第一个流形是1-连通的，第二个流形是测地线完备的.进一步证明了连通Banach对称空间$M$的自同构群可以转化为光滑传递作用在$M$上的Banach-李群.特别地，$M$是Banach齐性空间。

整体例外黎曼$4$ -对称空间$E_8/(E_8)^{σ'_{4}}$的实现

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

紧单连通黎曼4-对称空间是J.A.吉姆·内兹作为齐次流形，这些空间是$G/H$的，其中$G$是一个连通的紧致单李群，其自同构$\tildeγ$为4阶，$H$是$G$的一个不动点子群$G^γ$.本文对例外紧李群$G=E_8$，给出了由$\boldsymbol{R}$ -线性变换$σ'_{4}$诱导的4阶自同构的显式形式，并确定了群$(E_8)^{σ'_{4}}$的结构.从而实现了全局例外的黎曼$4$ -对称空间$E_8/(E_8)^{σ'_{4}}$ .

基回路群的实轨迹

Real loci of based loop groups

设$(G,K)$是最大秩的黎曼对称对，其中$G$是紧致单连通李群，$K$是对合自同构$σ$的不动点集.这导致了基循环空间$Ω(G)$的对合自同构$τ$。存在一个极大环面$T\subset G$，使得$T\times S^1$对$Ω(G)$的正则作用与$τ$相容（在Duistermaat意义上）。这使我们能够制定和证明一个版本的Duistermaat的凸性定理。即，在$Ω(G)$上的$T\times S^1$矩映射下的$Ω(G)$和$Ω(G)^τ$（$τ$的不动点集）的图像是相等的。空间$Ω(G)^τ$同伦等价于黎曼对称空间$G/K$的循环空间$Ω(G/K)$。我们证明了Bott和Samelson的一个结果的一个更强的形式，它将上同调环与$Ω(G)$和$Ω(G/K)$的$\mathbb{Z}_2$中的系数联系起来。也就是说，这两个上同调环是同构的，通过度减半同构（Bott和Samelson证明了Betti数相等）。这个定理的一个版本涉及等变上同调也被证明。证明使用的概念共轭空间的意义上的豪斯曼，霍尔姆，和Puppe。

On exceptional compact homogeneous geometries of type C3

We provide a uniform framework to study the exceptional homogeneous compact geometries of type C3. This framework is then used to show that these are simply connected, answering a question by Kramer and Lytchak, and to calculate the full automorphism groups.

On some realizations of globally exceptional $\varmathbb{Z}_3 \times \varmathbb{Z}_3 $-symmetric spaces $G/K$, $G=G_2, F_4, E_6$, Part I

R. Lutz introduced the notion of $\varGamma$-symmetric space as a generalization of the classical notion of symmetric space in 1981, where $\varGamma$ is a finite abelian group. In the present article, as $\varGamma=\varmathbb{Z}_3 \times \varmathbb{Z}_3$, we give the automorphisms $\tildeσ_3, \tildeτ_3$ of order $3$ on the connected compact exceptional Lie groups $G=G_2, F_4,E_6$ %and construct $\varGamma=\varmathbb{Z}_3 \times \varmathbb{Z}_3$ as the elements of order $3$ in $\Aut(G)$, explicitly and determine the structure of the group $G^{σ_3} \cap G^{τ_3}$ using homomorphism theorem eleme…

Exceptional complex structures and the hypermultiplet moduli of 5d Minkowski compactifications of M-theory

We present a detailed study of a new mathematical object in $\mathrm{E}_{6(6)}\times \mathbb{R}^{+}$ generalised geometry called an `exceptional complex structure' (ECS). It is the extension of a conventional complex structure to one that includes all the degrees of freedom of M-theory or type IIB supergravity in six or five dimensions, and as such characterises, in part, the geometry of generic supersymmetric compactifications to five-dimensional Minkowkski space. We define an ECS as an integrable $\mathrm{U}^{*}(6)\times \mathbb{R}^{+}$ structure and show it is equivalent to a particular for…

运动学李代数与辛对称空间 I：李代数方面

Kinematical Lie algebras and symplectic symmetric spaces I: Lie algebraic aspects

本文研究了广义运动学李代数的结构及其与辛对称空间的关系，提出了辛involution李代数的构造方法，并给出了其完整分类。

Almost complex structures in 6D with nondegenerate Nijenhuis tensors and large symmetry groups

For an almost complex structure $J$ in dimension 6 with nondegenerate Nijenhuis tensor $N_J$, the automorphism group $G=Aut(J)$ of maximal dimension is the exceptional Lie group $G_2$. In this paper we establish that the sub-maximal dimension of automorphism groups of almost complex structures with nondegenerate $N_J$, i.e. the largest realizable dimension that is less than 14, is $\dim G=10$. Next we prove that only 3 spaces realize this, and all of them are strictly nearly (pseudo-) Kähler and globally homogeneous. Moreover, we show that all examples with $\dim Aut(J)=9$ have semi-simple iso…