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On Homogeneous Spaces, Holonomy, and Non-Associative Algebras

Published online by Cambridge University Press:  22 January 2016

Arthur A. Sagle
Arthur A. Sagle*
Affiliation:
University of Minnesota
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Let G be a connected Lie group and H a closed subgroup. The homogeneous space M = G/H is called reductive if in the Lie algebra g of G there exists a subspace m such that (subspace direct sum) and where is the Lie algebra of H, see [8].

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 32 , June 1968 , pp. 373 - 394
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1968

References

[1] Helgason, S.: Differential Geometry and Symmetric Spaces, Academic Press, 1962.Google Scholar
[2] Jacobson, N.: Lie Algebras, John Wiley, 1962.Google Scholar
[3] Jacobson, N.: Structure of Rings, American Mathematical Society, 1956.Google Scholar
[4] Jacobson, N.: Completely reducible Lie algebras of linear transformations, Proc. Amer. Math. Soc., Vol. 2 (1951), 105–113.CrossRefGoogle Scholar
[5] Kostant, B.: On holonomy and homogeneous spaces, Nagoya Math. Jol., Vol. 12 (1957), 31–54.Google Scholar
[6] Loos, O.: Thesis, University of Munich, 1966.Google Scholar
[7] Nomizu, K.: Recent development in the theory of connections and holonomy groups, Advances in Mathematics, Vol. 1, Academic Press, 1961.Google Scholar
[8] Nomizu, K.: Invariant affine connections on homogeneous spaces, Amer. Math. J., Vol. 76 (1954), 33–65.Google Scholar
[9] Nomizu, K.: Studies on Riemannian homogeneous spaces, Nagoya Math. J., Vol.9 (1955), 43–56.Google Scholar
[10] Sagle, A.: On anti-commutative algebras and homogeneous spaces, to appear J. of Math, and Mech., 1967.Google Scholar
[11] Sagle, A.: A note on simple anti-commutative algebras obtained from reductive homogeneous spaces, to appear in Nagoya Math. J.CrossRefGoogle Scholar
[12] Sagle, A.: A note on triple systems and totally geodesic submanifolds in a homogeneous space, to appear in Nagoya Math. J.Google Scholar
[13] Sagle, A. and Winter, D.J.: On homogeneous spaces and reductive subalgebras of simple Lie algebras, to appear in Trans. Amer. Math. Soc.CrossRefGoogle Scholar
[14] Schafer, R.D.: Inner derivations of nonassociative algebras, Bull. Amer. Math. Soc., Vol. 55 (1949), 769–776.CrossRefGoogle Scholar
[15] Wolf, J.: Complex homogeneous contact manifolds and quaternionic symmetric spaces, J. of Math, and Mech., Vol. 14 (1965), 1033–1042.Google Scholar
[16] Wolf, J.: The geometry and structure of isotropy-irreducible homogeneous spaces, to appear.Google Scholar
[17] Yamaguti, K.: Note on Malcev algebras, Kumamoto J. Sci., Vol. 5 (1962), 171–184.Google Scholar
[18] Hano, J. and Matsushima, Y.: Some studies on Kaehlerian homogeneous spaces, Nagoya Math. J., Vol. 11 (1957), 77–92.Google Scholar