| Title:
|
Homogeneous Einstein manifolds based on symplectic triple systems (English) |
| Author:
|
Fontanals, Cristina Draper |
| Language:
|
English |
| Journal:
|
Communications in Mathematics |
| ISSN:
|
1804-1388 (print) |
| ISSN:
|
2336-1298 (online) |
| Volume:
|
28 |
| Issue:
|
2 |
| Year:
|
2020 |
| Pages:
|
139-154 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
For each simple symplectic triple system over the real numbers, the standard enveloping Lie algebra and the algebra of inner derivations of the triple provide a reductive pair related to a semi-Riemannian homogeneous manifold. It is proved that this is an Einstein manifold. (English) |
| Keyword:
|
Einstein metric |
| Keyword:
|
symplectic triple system |
| Keyword:
|
homogeneous manifold |
| Keyword:
|
curvature |
| Keyword:
|
3\discretionary-Sasakian manifold |
| Keyword:
|
Freudenthal triple system |
| MSC:
|
17A40 |
| MSC:
|
17B60 |
| MSC:
|
53C30 |
| MSC:
|
53C50 |
| idZBL:
|
Zbl 07300186 |
| idMR:
|
MR4162926 |
| . |
| Date available:
|
2021-03-03T08:44:58Z |
| Last updated:
|
2021-03-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148699 |
| . |
| Reference:
|
[1] Alekseevski, D.V.: Homogeneous Einstein metrics.Differential Geometry and its Applications (Proccedings of the Conference), 1987, 1-21, Univ. J. E. Purkyně, Brno, MR 0923361 |
| Reference:
|
[2] Alekseevsky, D.V., Cortés, V.: The twistor spaces of a para-quaternionic K{ä}hler manifold.Osaka Journal of Mathematics, 45, 1, 2008, 215-251, Osaka University and Osaka City University, Departments of Mathematics, MR 2416658 |
| Reference:
|
[3] Arvanitoyeorgos, A., Chrysikos, I.: Invariant Einstein metrics on generalized flag manifolds with two isotropy summands.Journal of the Australian Mathematical Society, 90, 2, 2011, 237-251, Cambridge University Press, MR 2821781, 10.1017/S1446788711001303 |
| Reference:
|
[4] Arvanitoyeorgos, A., Mori, K., Sakane, Y.: Einstein metrics on compact Lie groups which are not naturally reductive.Geometriae Dedicata, 160, 1, 2012, 261-285, Springer, MR 2970054, 10.1007/s10711-011-9681-1 |
| Reference:
|
[5] Arvanitoyeorgos, A., Sakane, Y., Statha, M.: New homogeneous Einstein metrics on quaternionic Stiefel manifolds.Advances in Geometry, 18, 4, 2018, 509-524, De Gruyter, MR 3871412 |
| Reference:
|
[6] Benito, P., Draper, C., Elduque, A.: Lie-Yamaguti algebras related to $\mathfrak {g}_2$.Journal of Pure and Applied Algebra, 202, 1-3, 2005, 22-54, Elsevier, MR 2163399, 10.1016/j.jpaa.2005.01.003 |
| Reference:
|
[7] Benito, P., Elduque, A., Martín-Herce, F.: Nonassociative systems and irreducible homogeneous spaces.Recent Advances in Geometry and Topology, 2004, 65-76, Cluj Univ. Press, Cluj-Napoca, MR 2113571 |
| Reference:
|
[8] Bertram, W.: The geometry of Jordan and Lie structures.1754, 2000, Springer-Verlag, Berlin, Lecture Notes in Mathematics 1754. MR 1809879 |
| Reference:
|
[9] Besse, A.L.: Einstein manifolds.2008, Classics in Mathematics. Springer-Verlag, Berlin, Reprint of the 1987 edition. MR 2371700 |
| Reference:
|
[10] Böhm, C., Kerr, M.M.: Low-dimensional homogeneous Einstein manifolds.Transactions of the American Mathematical Society, 4, 2006, 1455-1468, JSTOR, MR 2186982 |
| Reference:
|
[11] Boyer, C., Galicki, K.: Sasakian geometry.2008, Oxford Univ. Press, MR 2382957 |
| Reference:
|
[12] Dancer, A.S., Jørgensen, H.R., Swann, A.F.: Metric geometries over the split quaternions.Rendiconti del Seminario Matematico, Universit¸ e Politecnico di Torino, 63, 2, 2005, 119-139, MR 2143244 |
| Reference:
|
[13] C. Draper: Holonomy and 3-Sasakian homogeneous manifolds versus symplectic triple systems.Transformation Groups, 2019, arXiv:1903.07815. |
| Reference:
|
[14] C. Draper, A. Elduque: Classification of simple real symplectic triple systems.preprint. |
| Reference:
|
[15] C. Draper, M. Ortega, F.J. Palomo: Affine Connections on 3-Sasakian Homogeneous Manifolds.Mathematische Zeitschrift, 294, 2020, 817-868, MR 4054456, 10.1007/s00209-019-02304-x |
| Reference:
|
[16] Elduque, A.: New simple Lie superalgebras in characteristic 3.Journal of Algebra, 296, 1, 2006, 196-233, Elsevier, MR 2192604, 10.1016/j.jalgebra.2005.06.014 |
| Reference:
|
[17] Elduque, A.: The Magic Square and Symmetric Compositions II.Revista Matemática Iberoamericana, 23, 1, 2007, 57-84, Departamento de Matemáticas, Universidad Aut{ó}noma de Madrid, MR 2351126 |
| Reference:
|
[18] Heber, J.: Noncompact homogeneous Einstein spaces.Inventiones mathematicae, 133, 2, 1998, 279-352, Springer, MR 1632782, 10.1007/s002220050247 |
| Reference:
|
[19] Kashiwada, T.: A note on a Riemannian space with Sasakian 3-structure.Natural Science Report, Ochanomizu University, 22, 1, 1971, 1-2, MR 0303449 |
| Reference:
|
[20] Kerner, R.: Ternary and non-associative structures.International Journal of Geometric Methods in Modern Physics, 5, 8, 2008, 1265-1294, World Scientific, MR 2484553 |
| Reference:
|
[21] Meyberg, K.E.: Eine Theorie der Freudenthalschen Tripelsysteme I, II (German).Koninklijke Nederlandse Akademie van Wetenschappen Proceedings. Series A = Indagationes Mathematicae, 30, 1968, 162-190, MR 0225838, 10.1016/S1385-7258(68)50018-0 |
| Reference:
|
[22] Tamaru, H.: Parabolic subgroups of semisimple Lie groups and Einstein solvmanifolds.Mathematische Annalen, 351, 1, 2011, 51-66, Springer, MR 2824845, 10.1007/s00208-010-0589-0 |
| Reference:
|
[23] Tamaru, H.: Noncompact homogeneous Einstein manifolds attached to graded Lie algebras.Mathematische Zeitschrift, 259, 1, 2008, 171-186, Springer, MR 2377747, 10.1007/s00209-007-0217-1 |
| Reference:
|
[24] M.Y. Wang, W. Ziller: On normal homogeneous Einstein manifolds.Annales Scientifiques de l'Ecole Normale Sup{é}rieure, 18, 4, 1985, 563-633, Zbl 0598.53049, MR 0839687, 10.24033/asens.1497 |
| Reference:
|
[25] M.Y. Wang, W. Ziller: Existence and non-existence of homogeneous Einstein metrics.Inventiones Mathematicae, 84, 1, 1986, 177-194, Springer-Verlag, MR 0830044, 10.1007/BF01388738 |
| Reference:
|
[26] Wolf, J.A.: The geometry and structure of isotropy irreducible homogeneous spaces.Acta Mathematica, 120, 1968, 59-148, Institut Mittag-Leffler, MR 0223501 |
| Reference:
|
[27] Yamaguti, K., Asano, H.: On the Freudenthal's construction of exceptional Lie algebras.Proceedings of the Japan Academy, 51, 4, 1975, 253-258, The Japan Academy, MR 0374212, 10.3792/pja/1195518629 |
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