Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A solution of the problem of standard compact Clifford-Klein forms

Maciej Bochenski, Aleksy Tralle

TL;DR

The article resolves the classification of standard compact Clifford-Klein forms for homogeneous spaces $G/H$ in the setting where $G$ is absolutely simple and $H,L$ are reductive, under mild extra hypotheses on the embeddings. It proves that such standard forms arise exclusively from Lie algebra decompositions with $ rak g= rak h+ rak l$ and compact intersection $ rak higcap rak l$, by combining root-system methods, well-embeddedness concepts, and Dynkin classification to rule out non-decomposable cases. A key outcome is a practical criterion: a reductive subgroup $L$ acts properly and cocompactly on $G/H$ if and only if $G=H\,L$ and $Higcap L$ is compact. The results advance Kobayashi’s program by clarifying when standard compact Clifford-Klein forms exist and by providing explicit structural conditions that guarantee (or preclude) proper cocompact actions in this important class of homogeneous spaces.

Abstract

We solve the long standing problem of classification of standard compact Clifford-Klein forms of homogeneous spaces of simple non-compact real Lie groups under the extra assumption that $G$, $H$, $L$ are simple and absolutely simple. Then the result is that standard compact Clifford-Klein forms always arise from triples $(\mathfrak{g},\mathfrak{h},\mathfrak{l})$ of real Lie algebras such that $\mathfrak{h}\subset\mathfrak{g},\mathfrak{l}\subset\mathfrak{g}$, $\mathfrak{g}$ is simple and absolutely simple, $\mathfrak{h},\mathfrak{l}$ are (non-compact) reductive, $\mathfrak{g}=\mathfrak{h}+\mathfrak{l}$, and the intersection $\mathfrak{h}\cap\mathfrak{l}$ is compact. The consequence of this is the following characterization of proper co-compact actions of reductive Lie subgroups $L\subset G$ on a homogeneous spaces $G/H$ determined by absolutely simple real Lie group $G$ and a closed reductive subgroup $H$: $L$ acts on $G/H$ properly and co-compactly if and only if $G=H\cdot L$ and $H\cap L$ is compact.

A solution of the problem of standard compact Clifford-Klein forms

TL;DR

The article resolves the classification of standard compact Clifford-Klein forms for homogeneous spaces in the setting where is absolutely simple and are reductive, under mild extra hypotheses on the embeddings. It proves that such standard forms arise exclusively from Lie algebra decompositions with and compact intersection , by combining root-system methods, well-embeddedness concepts, and Dynkin classification to rule out non-decomposable cases. A key outcome is a practical criterion: a reductive subgroup acts properly and cocompactly on if and only if and is compact. The results advance Kobayashi’s program by clarifying when standard compact Clifford-Klein forms exist and by providing explicit structural conditions that guarantee (or preclude) proper cocompact actions in this important class of homogeneous spaces.

Abstract

We solve the long standing problem of classification of standard compact Clifford-Klein forms of homogeneous spaces of simple non-compact real Lie groups under the extra assumption that , , are simple and absolutely simple. Then the result is that standard compact Clifford-Klein forms always arise from triples of real Lie algebras such that , is simple and absolutely simple, are (non-compact) reductive, , and the intersection is compact. The consequence of this is the following characterization of proper co-compact actions of reductive Lie subgroups on a homogeneous spaces determined by absolutely simple real Lie group and a closed reductive subgroup : acts on properly and co-compactly if and only if and is compact.
Paper Structure (20 sections, 25 theorems, 69 equations, 4 tables)

This paper contains 20 sections, 25 theorems, 69 equations, 4 tables.

Table of Contents

  1. Introduction
  2. The problem and motivation
  3. Methods of proof of the main results
  4. Preliminaries
  5. Notation
  6. Properties of Lie algebra embeddings yielding compact Clifford-Klein forms
  7. Some properties of root systems
  8. Proof of Theorem \ref{['thm:class']}: eliminating the regular case
  9. Assumptions
  10. Plan of the proof
  11. Deriving the contradiction between 1) and 2)
  12. Constructing $Y\in\mathfrak{c},Y\not\in\mathfrak{m}_0,Y\in\mathfrak{z}_{\mathfrak{g}}(\mathfrak{h}),[Y,iX]=0$
  13. The structure of $\mathfrak{z}_{\mathfrak{g}^c}(\mathfrak{a}_{h})$
  14. Construction of $Y$
  15. Completion of proof of Theorem \ref{['thm:class']}: non-regular case
  16. ...and 5 more sections

Key Result

Theorem 1

Assume that $\mathfrak{g}$ is absolutely simple, and $\mathfrak{h}$ and $\mathfrak{l}$ are reductive. All triples $(\mathfrak{g},\mathfrak{h},\mathfrak{l})$ listed in Table 1 yield standard compact Clifford-Klein forms.

Theorems & Definitions (42)

  • Definition 1: T. Kobayashi, k-k,kob
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 1
  • Lemma 1: botr
  • Definition 2
  • Proposition 1: botr
  • Theorem 4: botr
  • Corollary 1: botr
  • ...and 32 more