Semisimple subalgebras in simple Lie algebras and a computational approach to the compact Clifford-Klein forms problem

TL;DR

This paper introduces algorithms and a computer program to identify homogeneous spaces of semisimple non-compact Lie groups that lack compact Clifford-Klein forms, based on cohomological obstructions.

Contribution

It presents a computational approach to determine the existence of compact Clifford-Klein forms using cohomological criteria, advancing the study of homogeneous spaces.

Findings

Large class of spaces satisfy Tholozan's obstruction

Algorithm effectively identifies non-existence cases

Computational method complements theoretical analysis

Abstract

In this paper we develop algorithms of finding homogeneous spaces of semisimple non-compact Lie groups which do not admit compact Clifford-Klein forms. We propose a computer program which checks if the given homogeneous space has a non-vanishing cohomological obstruction (found by Tholozan) to compact Clifford-Klein forms. By a numerical experiment we show that there is a large class of homogeneous spaces satisfying Tholozan's condition.