Weakly irreducible subgroups of $Sp(1,n+1)$

Connected H-irreducible Lie subgroups of Sp(1,n)

In this article we classify all connected H-irreducible Lie subgroups of Sp(1,n) up to conjugacy.

On weakly reflective submanifolds in compact isotropy irreducible Riemannian homogeneous spaces

We show that for any weakly reflective submanifold of a compact isotropy irreducible Riemannian homogeneous space its inverse image under the parallel transport map is an infinite dimensional weakly reflective PF submanifold of a Hilbert space. This is an extension of the author's previous result in the case of compact irreducible Riemannian symmetric spaces. We also give a characterization of so obtained weakly reflective PF submanifolds.

Lightlike foliations on Lorentzian manifolds with weakly irreducible holonomy algebra

We study the lightlike foliations that appear on Lorentzian manifolds with weakly irreducible not irreducible holonomy algebra. We give global structure equations for the foliation that generalize the Gauss and Weingarten equations for one lightlike hypersurface. This gives us some global operators on the manifold. Using these operators, we decompose the curvature tensor of the manifold into several components. We give a criteria how to find the type of the holonomy algebras (there are 4 possible types) in terms of the global operators.

弱稳定不可约杨-米尔斯场在$S^4$上的研究

Weakly stable irreducible Yang-Mills fields over $S^4$

本文研究了在$S^4$上弱稳定不可约杨-米尔斯场的自对偶性，证明了在简单李代数结构下，任何弱稳定不可约连接必为自对偶或反自对偶，并指出若李代数有非平凡阿贝尔中心，则不存在不可约杨-米尔斯场。

A note on trace fields of complex hyperbolic groups

We show that if $Γ$ is an irreducible subgroup of ${\rm SU}(2,1)$, then $Γ$ contains a loxodromic element $A$. If $A$ has eigenvalues $λ_1 = λe^{iφ},$ $λ_2 = e^{-2iφ}$, $λ_3 = λ^{-1}e^{iφ}$, we prove that $Γ$ is conjugate in ${\rm SU}(2,1)$ to a subgroup of ${\rm SU}(2,1,\mathbb{Q}(Γ,λ)),$ where $\mathbb{Q}(Γ, λ)$ is the field generated by the trace field $\mathbb{Q}(Γ)$ of $Γ$ and $λ$. It follows from this that if $Γ$ is an irreducible subgroup of ${\rm SU}(2,1)$ such that the trace field $\mathbb{Q}(Γ)$ is real, then $Γ$ is conjugate in ${\rm SU}(2,1)$ to a subgroup of ${\rm SO}(2,1)$. As a …

Holonomy groups and special geometric structures of pseudo-Kählerian manifolds of index 2

The problem of classification of connected holonomy groups (equivalently of holonomy algebras) for pseudo-Riemannian manifolds is open. The classification of Riemannian holonomy algebras is a classical result. The classification of Lorentzian holonomy algebras was obtained recently. In the present paper weakly-irreducible not irreducible subalgebras of $\su(1,n+1)$ ($n\geq 0$) are classified. Weakly-irreducible not irreducible holonomy algebras of pseudo-Kählerian and special pseudo-Kählerian manifolds are classified. An example of metric for each possible holonomy algebra is given. This gives…