Reductive homogeneous spaces and nonassociative algebras

Elduque, Alberto

Communications in Mathematics, Tome 28 (2020) no. 2, pp. 199-229

Abstract (VO)

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.

Détail

Classification : 17A99, 17B60, 22F30, 53B05, 53C30
Keywords: Reductive homogeneous space; invariant affine connection; Lie-Yamaguti algebra

Bibliographie

[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

[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, | DOI | MR

[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, | DOI | MR

[4] Benito, P., Elduque, A., Martín-Herce, F.: Irreducible Lie-Yamaguti algebras. Journal of Pure and Applied Algebra, 213, 5, 2009, 795-808, Elsevier, | DOI | MR

[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, | DOI | MR

[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, | DOI | MR

[7] Conlon, L.: Differentiable manifolds, Second edition. 2001, Birkhäuser, Boston, MA, | MR

[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, | DOI | MR

[9] Elduque, A., Kochetov, M.: Gradings on simple Lie algebras. 189, 2013, American Mathematical Soc., | MR

[10] Elduque, A., Myung, H.C.: Color algebras and affine connections on $S^6$. Journal of Algebra, 149, 1, 1992, 234-261, Academic Press, | DOI | MR

[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, | DOI | MR

[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

[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, | DOI | MR

[14] Kobayashi, S., Nomizu, K.: Foundations of differential geometry I. 1, 2, 1963, Interscience Publishers, New York-London, | MR | Zbl

[15] Kobayashi, S., Nomizu, K.: Foundations of differential geometry II. 1, 2, 1969, Interscience Publishers, New York-London-Sidney, | MR

[16] Laquer, H.T.: Invariant affine connections on Lie groups. Transactions of the American Mathematical Society, 331, 2, 1992, 541-551, JSTOR, | DOI | MR

[17] Laquer, H.T.: Invariant affine connections on symmetric spaces. Proceedings of the American Mathematical Society, 1992, 447-454, JSTOR, | DOI | MR

[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

[19] Myung, H.C.: Malcev-admissible algebras. 1986, Birkhäuser Boston, Inc., Boston, MA, Progress in Mathematics, 64. | MR

[20] Nagai, S.: The classification of naturally reductive homogeneous real hypersurfaces in complex projective space. Archiv der Mathematik, 69, 6, 1997, 523-528, Springer, | DOI | MR

[21] Nomizu, K.: Invariant affine connections on homogeneous spaces. American Journal of Mathematics, 76, 1, 1954, 33-65, JSTOR, | DOI | MR

[22] Sagle, A.A.: Jordan algebras and connections on homogeneous spaces. Transactions of the American Mathematical Society, 187, 1974, 405-427, | DOI | MR

[23] Vinberg, E.B.: Invariant linear connections in a homogeneous space. Trudy Moskovskogo Matematicheskogo Obshchestva, 9, 1960, 191-210, Moscow Mathematical Society, | MR

[24] Wang, H.C.: On invariant connections over a principal fibre bundle. Nagoya Mathematical Journal, 13, 1958, 1-19, Cambridge University Press, | DOI | MR

[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

[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