×

Nonexistence of standard compact Clifford-Klein forms of homogeneous spaces of exceptional Lie groups. (English) Zbl 1432.57065

Compact Clifford-Klein forms of homogeneous spaces of simple exceptional Lie groups are considered. There is one special construction of Clifford-Klein forms. Let \(G\) be a semisimple linear real Lie group. Let us suppose that there exists a reductive noncompact Lie subgroup \(L \subset G\) such that \(L\) acts properly on \(G/H\) and \(L \setminus G/H\) is compact. Then for any co-compact torsion-free lattice \(\Gamma \subset L\) the space \(\Gamma \setminus G/H\) is the compact Clifford-Klein form, which called standard Clifford-Klein form for \(G/H\).
It is proved in this article that if \(G\) is a simple linear real Lie group of exceptional type, \(H \subset G\) is a reductive non-compact subgroup such that \(G/H\) is noncompact, then \(G/H\) does not admit standard compact Clifford-Klein forms. It is dome with using computer algebra system GAP and some special algorithms of classifying semisimple subalgebras in simple Lie algebras and some databases. The authors created a special plugin CKForms, avaiable online, which uses two plugins – SLA and CoReLG. Also Kobayashi’s criterion for properness of the Lie group action is used.

MSC:

57S30 Discontinuous groups of transformations
22F30 Homogeneous spaces
22E40 Discrete subgroups of Lie groups
17B20 Simple, semisimple, reductive (super)algebras
22-08 Computational methods for problems pertaining to topological groups

Software:

CKForms; GAP; SLA; CoReLG
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Benoist, Yves; Labourie, Fran\c{c}ois, Sur les espaces homog\`enes mod\`eles de vari\'{e}t\'{e}s compactes, Inst. Hautes \'{E}tudes Sci. Publ. Math., 76, 99-109 (1992) · Zbl 0786.53031
[2] bjstw M. Boche\'nski, P. Jastrz\c ebski, A. Szczepkowska, A. Tralle, and A. Woike, Semisimple subalgebras in simple Lie algebras and a computational approach to the compact Clifford-Klein forms problem, Experimental Mathematics, https://doi.org/10.1080/10586458.2018.1492475. · Zbl 1509.22012
[3] ckforms M. Boche\'nski, P. Jastrz\c ebski, and A. Tralle, CKForms. A GAP Package. Avaiable online (https://pjastr.github.io/CKForms/). · Zbl 1492.22008
[4] Boche\'{n}ski, Maciej; Jastrz\polhk ebski, Piotr; Okuda, Takayuki; Tralle, Aleksy, Proper \(SL(2,\mathbb{R})\)-actions on homogeneous spaces, Internat. J. Math., 27, 13, 1650106, 10 pp. (2016) · Zbl 1358.57035
[5] Boche\'{n}ski, Maciej; Ogryzek, Marek, A restriction on proper group actions on homogeneous spaces of reductive type, Geom. Dedicata, 178, 405-411 (2015) · Zbl 1329.57038
[6] Boche\'{n}ski, Maciej; Tralle, Aleksy, Clifford-Klein forms and a-hyperbolic rank, Int. Math. Res. Not. IMRN, 15, 6267-6285 (2015) · Zbl 1327.53063
[7] Boche\'{n}ski, Maciej; Tralle, Aleksy, On solvable compact Clifford-Klein forms, Proc. Amer. Math. Soc., 145, 4, 1819-1832 (2017) · Zbl 1359.22012
[8] Calabi, E.; Markus, L., Relativistic space forms, Ann. of Math. (2), 75, 63-76 (1962) · Zbl 0101.21804
[9] Dietrich, Heiko; Faccin, Paolo; de Graaf, Willem A., Computing with real Lie algebras: real forms, Cartan decompositions, and Cartan subalgebras, J. Symbolic Comput., 56, 27-45 (2013) · Zbl 1332.17001
[10] Dynkin, E. B., Semisimple subalgebras of semisimple Lie algebras, Mat. Sbornik N.S., 30(72), 349-462 (3 plates) (1952) · Zbl 0048.01701
[11] Faccin, Paolo; de Graaf, Willem A., Constructing Semisimple Subalgebras of Real Semisimple Lie Algebras. Lie algebras and related topics, Contemp. Math. 652, 75-89 (2015), Amer. Math. Soc., Providence, RI · Zbl 1367.17008
[12] de Graaf, Willem A., Constructing semisimple subalgebras of semisimple Lie algebras, J. Algebra, 325, 416-430 (2011) · Zbl 1255.17007
[13] sla W. A. de Graaf, SLA-Computing with Simple Lie Algebras, A GAP Package. Available online (http://www.science.unitn.it/ degraaf/sla.html).
[14] gap The GAP Group, GAP-Groups, Algorithms and Programming, v.4.10. Available online (https://www.gap-system.org/).
[15] Helgason, Sigurdur, Differential Geometry, Lie Groups, and Symmetric Spaces, Pure and Applied Mathematics 80, xv+628 pp. (1978), Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London · Zbl 0451.53038
[16] Kassel, Fanny, Deformation of proper actions on reductive homogeneous spaces, Math. Ann., 353, 2, 599-632 (2012) · Zbl 1248.22005
[17] Kassel, Fanny; Kobayashi, Toshiyuki, Stable spectrum for pseudo-Riemannian locally symmetric spaces, C. R. Math. Acad. Sci. Paris, 349, 1-2, 29-33 (2011) · Zbl 1208.22013
[18] Kobayashi, Toshiyuki; Yoshino, Taro, Compact Clifford-Klein forms of symmetric spaces-revisited, Pure Appl. Math. Q., 1, 3, Special Issue: In memory of Armand Borel., 591-663 (2005) · Zbl 1145.22011
[19] Kobayashi, Toshiyuki, Deformation of compact Clifford-Klein forms of indefinite-Riemannian homogeneous manifolds, Math. Ann., 310, 3, 395-409 (1998) · Zbl 0891.22014
[20] Kobayashi, Toshiyuki, Proper action on a homogeneous space of reductive type, Math. Ann., 285, 2, 249-263 (1989) · Zbl 0662.22008
[21] Kobayashi, Toshiyuki, A necessary condition for the existence of compact Clifford-Klein forms of homogeneous spaces of reductive type, Duke Math. J., 67, 3, 653-664 (1992) · Zbl 0799.53056
[22] Kobayashi, Toshiyuki, Discontinuous groups acting on homogeneous spaces of reductive type. Representation theory of Lie groups and Lie algebras, Fuji-Kawaguchiko, 1990, 59-75 (1992), World Sci. Publ., River Edge, NJ · Zbl 1193.22010
[23] Kobayashi, Toshiyuki, Discontinuous groups and Clifford-Klein forms of pseudo-Riemannian homogeneous manifolds. Algebraic and analytic methods in representation theory, S\o nderborg, 1994, Perspect. Math. 17, 99-165 (1997), Academic Press, San Diego, CA · Zbl 0899.43005
[24] Kobayashi, Toshiyuki; Ono, Kaoru, Note on Hirzebruch’s proportionality principle, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 37, 1, 71-87 (1990) · Zbl 0726.57019
[25] Kulkarni, Ravi S., Proper actions and pseudo-Riemannian space forms, Adv. in Math., 40, 1, 10-51 (1981) · Zbl 0462.53041
[26] Margulis, Gregory, Existence of compact quotients of homogeneous spaces, measurably proper actions, and decay of matrix coefficients, Bull. Soc. Math. France, 125, 3, 447-456 (1997) · Zbl 0892.22009
[27] Minchenko, A. N., Semisimple subalgebras of exceptional Lie algebras, Tr. Mosk. Mat. Obs.. Trans. Moscow Math. Soc., 67, 225-259 (2006) · Zbl 1152.17003
[28] Morita, Yosuke, A topological necessary condition for the existence of compact Clifford-Klein forms, J. Differential Geom., 100, 3, 533-545 (2015) · Zbl 1323.53056
[29] Onishchik, Arkady L., Lectures on Real Semisimple Lie Algebras and Their Representations, ESI Lectures in Mathematics and Physics, x+86 pp. (2004), European Mathematical Society (EMS), Z\"{u}rich · Zbl 1080.17001
[30] ov A. L. Onishchik and E. B. Vinberg, Lie Groups and Lie Algebras III, Springer, Berlin, 2002. · Zbl 0797.22001
[31] Oh, Hee; Witte, Dave, Compact Clifford-Klein forms of homogeneous spaces of \(\text{SO}(2,n)\), Geom. Dedicata, 89, 25-57 (2002) · Zbl 1002.57074
[32] Okuda, Takayuki, Classification of semisimple symmetric spaces with proper \(SL(2,\mathbb{R})\)-actions, J. Differential Geom., 94, 2, 301-342 (2013) · Zbl 1312.53076
[33] Th N. Tholozan, Volume and non-exisyence of compact Clifford-Klein forms, ArXiv: 1511.09448.
[34] Tojo, Koichi, Classification of irreducible symmetric spaces which admit standard compact Clifford-Klein forms, Proc. Japan Acad. Ser. A Math. Sci., 95, 2, 11-15 (2019) · Zbl 1416.53050
[35] Zimmer, Robert J., Discrete groups and non-Riemannian homogeneous spaces, J. Amer. Math. Soc., 7, 1, 159-168 (1994) · Zbl 0801.22009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.