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

Abstract

In this article, we develop the algorithm for 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.

Keywords:

• semisimple subalgebra
• Clifford–Klein form

MATHEMATICS SUBJECT CLASSIFICATION:

• 17B20
• 22F30
• 22E40
• 65-05
• 65F

Acknowledgments

We thank Willem de Graaf for his help in getting some literature sources.