| Title:
|
Reductive homogeneous spaces and nonassociative algebras (English) |
| Author:
|
Elduque, Alberto |
| Language:
|
English |
| Journal:
|
Communications in Mathematics |
| ISSN:
|
1804-1388 (print) |
| ISSN:
|
2336-1298 (online) |
| Volume:
|
28 |
| Issue:
|
2 |
| Year:
|
2020 |
| Pages:
|
199-229 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The purpose of these survey notes is to give a presentation of a classical theorem of Nomizu \cite {Nom54} that relates the invariant affine connections on reductive homogeneous spaces and nonassociative algebras. (English) |
| Keyword:
|
Reductive homogeneous space |
| Keyword:
|
invariant affine connection |
| Keyword:
|
Lie-Yamaguti algebra |
| MSC:
|
17A99 |
| MSC:
|
17B60 |
| MSC:
|
22F30 |
| MSC:
|
53B05 |
| MSC:
|
53C30 |
| idZBL:
|
Zbl 07300190 |
| idMR:
|
MR4162930 |
| . |
| Date available:
|
2021-03-03T08:51:46Z |
| Last updated:
|
2021-03-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148703 |
| . |
| Reference:
|
[1] Alekseevskiĭ, D.V., Gamkrelidze, R.V., Lychagin, V.V., Vinogradov, A.M.: Geometry I: Basic ideas and concepts of differential geometry.1, 1991, Springer, Berlin, Encyclopaedia Math. Sci. 28. MR 1300019 |
| Reference:
|
[2] Benito, P., Draper, C., Elduque, A.: On some algebras related to simple Lie triple systems.Journal of Algebra, 219, 1, 1999, 234-254, Elsevier, MR 1707670, 10.1006/jabr.1999.7877 |
| Reference:
|
[3] 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:
|
[4] Benito, P., Elduque, A., Martín-Herce, F.: Irreducible Lie-Yamaguti algebras.Journal of Pure and Applied Algebra, 213, 5, 2009, 795-808, Elsevier, MR 2494372, 10.1016/j.jpaa.2008.09.003 |
| Reference:
|
[5] Benito, P., Elduque, A., Martín-Herce, F.: Irreducible Lie-Yamaguti algebras of generic type.Journal of Pure and Applied Algebra, 215, 2, 2011, 108-130, Elsevier, MR 2720677, 10.1016/j.jpaa.2010.04.003 |
| Reference:
|
[6] Burde, D.: Left-symmetric algebras, or pre-Lie algebras in geometry and physics.Central European Journal of Mathematics, 4, 3, 2006, 323-357, Springer, MR 2233854, 10.2478/s11533-006-0014-9 |
| Reference:
|
[7] Conlon, L.: Differentiable manifolds, Second edition.2001, Birkhäuser, Boston, MA, MR 1821549 |
| Reference:
|
[8] Draper, C., Garvín, A., Palomo, F.J.: Invariant affine connections on odd-dimensional spheres.Annals of Global Analysis and Geometry, 49, 3, 2016, 213-251, Springer, MR 3485984, 10.1007/s10455-015-9489-6 |
| Reference:
|
[9] Elduque, A., Kochetov, M.: Gradings on simple Lie algebras.189, 2013, American Mathematical Soc., MR 3087174 |
| Reference:
|
[10] Elduque, A., Myung, H.C.: Color algebras and affine connections on $S^6$.Journal of Algebra, 149, 1, 1992, 234-261, Academic Press, MR 1165209, 10.1016/0021-8693(92)90014-D |
| Reference:
|
[11] Elduque, A., Myung, H.C.: The reductive pair $(B_3,G_2)$ and affine connections on $S^7$.Journal of Pure and Applied Algebra, 86, 2, 1993, 155-171, Elsevier, MR 1215643, 10.1016/0022-4049(93)90100-8 |
| Reference:
|
[12] Elduque, A., Myung, H.C.: The reductive pair $(B_{4}, B_{3})$ and affine connections on $S^{15}$.Journal of Algebra, 227, 2, 2000, 504-531, Academic Press, MR 1759833 |
| Reference:
|
[13] Kinyon, M.K., Weinstein, A.: Leibniz algebras, Courant algebroids, and multiplications on reductive homogeneous spaces.American Journal of Mathematics, 123, 3, 2001, 525-550, Johns Hopkins University Press, MR 1833152, 10.1353/ajm.2001.0017 |
| Reference:
|
[14] Kobayashi, S., Nomizu, K.: Foundations of differential geometry I.1, 2, 1963, Interscience Publishers, New York-London, Zbl 0119.37502, MR 0152974 |
| Reference:
|
[15] Kobayashi, S., Nomizu, K.: Foundations of differential geometry II.1, 2, 1969, Interscience Publishers, New York-London-Sidney, MR 0238225 |
| Reference:
|
[16] Laquer, H.T.: Invariant affine connections on Lie groups.Transactions of the American Mathematical Society, 331, 2, 1992, 541-551, JSTOR, MR 1075384, 10.1090/S0002-9947-1992-1075384-4 |
| Reference:
|
[17] Laquer, H.T.: Invariant affine connections on symmetric spaces.Proceedings of the American Mathematical Society, 1992, 447-454, JSTOR, MR 1107273, 10.1090/S0002-9939-1992-1107273-6 |
| Reference:
|
[18] Medina, P.A.: Flat left-invariant connections adapted to the automorphism structure of a Lie group.Journal of Differential Geometry, 16, 3, 1981, 445-474, Lehigh University, MR 0654637 |
| Reference:
|
[19] Myung, H.C.: Malcev-admissible algebras.1986, Birkhäuser Boston, Inc., Boston, MA, Progress in Mathematics, 64. MR 0885089 |
| Reference:
|
[20] Nagai, S.: The classification of naturally reductive homogeneous real hypersurfaces in complex projective space.Archiv der Mathematik, 69, 6, 1997, 523-528, Springer, MR 1480520, 10.1007/s000130050155 |
| Reference:
|
[21] Nomizu, K.: Invariant affine connections on homogeneous spaces.American Journal of Mathematics, 76, 1, 1954, 33-65, JSTOR, MR 0059050, 10.2307/2372398 |
| Reference:
|
[22] Sagle, A.A.: Jordan algebras and connections on homogeneous spaces.Transactions of the American Mathematical Society, 187, 1974, 405-427, MR 0339013, 10.1090/S0002-9947-1974-0339013-6 |
| Reference:
|
[23] Vinberg, E.B.: Invariant linear connections in a homogeneous space.Trudy Moskovskogo Matematicheskogo Obshchestva, 9, 1960, 191-210, Moscow Mathematical Society, MR 0176418 |
| Reference:
|
[24] Wang, H.C.: On invariant connections over a principal fibre bundle.Nagoya Mathematical Journal, 13, 1958, 1-19, Cambridge University Press, MR 0107276, 10.1017/S0027763000023461 |
| Reference:
|
[25] Warner, F.W.: Foundations of differentiable manifolds and Lie groups.1983, Springer-Verlag, New York-Berlin, Corrected reprint of the 1971 edition. Graduate Texts in Mathematics 94. MR 0722297 |
| Reference:
|
[26] Yamaguti, K.: On the Lie triple system and its generalization.Journal of Science of the Hiroshima University, Series A (Mathematics, Physics, Chemistry), 21, 3, 1958, 155-160, Hiroshima University, Department of Mathematics, MR 0100047 |
| . |
