2024/03/03 by Maciej Bocheński, Aleksy Tralle, Bochenski, Maciej +1
Mathematics · #Advanced Algebra and Geometry #Algebraic and Geometric Analysis #Differential Geometry (math.DG) #FOS: Mathematics #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2403.10539
openalex publication_date 2024/03/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 (\mathfrakg,\mathfrakh,\mathfrakl) of real Lie algebras such that \mathfrakh⊂\mathfrakg,\mathfrakl⊂\mathfrakg, \mathfrakg is simple and absolutely simple, \mathfrakh,\mathfrakl are (non-compact) reductive, \mathfrakg=\mathfrakh+\mathfrakl, and the intersection \mathfrakh∩\mathfrakl is compact. The consequence of this is the following characterization of proper co-compact actions of reductive Lie subgroups L⊂ 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⋅ L and H∩ L is compact.