Standard compact Clifford-Klein forms and Lie algebra decompositions

TL;DR

This paper explores the structure of Lie algebras related to standard compact Clifford-Klein forms, revealing new classes of homogeneous spaces that do not admit such forms, especially focusing on R-regular subalgebras.

Contribution

It establishes relations between root decompositions of Lie algebras and identifies conditions preventing the existence of standard compact Clifford-Klein forms.

Findings

Proper R-regular subalgebras do not generate homogeneous spaces with compact standard Clifford-Klein forms.

New classes of homogeneous spaces are identified that lack standard compact Clifford-Klein forms.

Relations between Lie algebra decompositions and Clifford-Klein forms are elucidated.

Abstract

We find relations between real root decompositions of triples of Lie algebras corresponding to standard compact Clifford-Klein forms, under the assumption that these triples are not Lie algebra decompositions in the sense of Onishchik. This enables us to find new classes of homogeneous spaces of simple real Lie groups which do not admit standard compact Clifford-Klein forms. In particular, we show that proper R-regular subalgebras of simple real Lie algebras never generate homogeneous spaces which admit compact standard Cliffrod-Klein forms.