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Nonexistence of standard compact Clifford-Klein forms of homogeneous spaces of exceptional Lie groups


Authors: Maciej Bocheński, Piotr Jastrzȩbski and Aleksy Tralle
Journal: Math. Comp. 89 (2020), 1487-1499
MSC (2010): Primary 57S30, 17B20, 22F30, 22E40, 65-05, 65FXX
DOI: https://doi.org/10.1090/mcom/3493
Published electronically: November 26, 2019
MathSciNet review: 4063325
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Abstract: We use a computer-aided approach to prove that there are no standard compact Clifford-Klein forms of homogeneous spaces of exceptional Lie groups. This yields further support for Kobayashi's conjecture about possible compact Clifford-Klein forms. Our approach is based on the algorithms developed in this work. They are inspired by the algorithmic methods of classifying semisimple subalgebras in simple Lie algebras developed by Faccin and de Graaf. Also, we use the databases created by them. These algorithms eliminate the majority of possibilities. We complete the proof using Kobayashi's criterion for properness of the Lie group action.


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Additional Information

Maciej Bocheński
Affiliation: Faculty of Mathematics and Computer Science, University of Warmia and Mazury, Słoneczna 54, 10-710 Olsztyn, Poland
Email: mabo@matman.uwm.edu.pl

Piotr Jastrzȩbski
Affiliation: Faculty of Mathematics and Computer Science, University of Warmia and Mazury, Słoneczna 54, 10-710 Olsztyn, Poland
Email: piojas@matman.uwm.edu.pl

Aleksy Tralle
Affiliation: Faculty of Mathematics and Computer Science, University of Warmia and Mazury, Słoneczna 54, 10-710 Olsztyn, Poland
Email: tralle@matman.uwm.edu.pl

DOI: https://doi.org/10.1090/mcom/3493
Keywords: Proper actions, semisimple algebras, Clifford--Klein forms
Received by editor(s): January 14, 2019
Received by editor(s) in revised form: July 1, 2019, and July 28, 2019
Published electronically: November 26, 2019
Additional Notes: The first author acknowledges the support of the National Science Center (grant NCN no. 2018/31/D/ST1/00083).
The third author acknowledges the support of the National Science Center (grant NCN no. 2018/31/B/ST1/00053).
Article copyright: © Copyright 2019 American Mathematical Society