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Transactions of the American Mathematical Society

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Jordan algebras and connections on homogeneous spaces


Author: Arthur A. Sagle
Journal: Trans. Amer. Math. Soc. 187 (1974), 405-427
MSC: Primary 53C30
DOI: https://doi.org/10.1090/S0002-9947-1974-0339013-6
MathSciNet review: 0339013
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Abstract: We use the correspondence between G-invariant connections on a reductive homogeneous space $ G/H$ and certain nonassociative algebras to explicitly compute the pseudo-Riemannian connections in terms of a Jordan algebra J of endomorphisms. It is shown that if G and H are semisimple Lie groups, then J is a semisimple Jordan algebra. Also a general method for computing examples of J is given.


References [Enhancements On Off] (What's this?)

  • [1] E. B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Mat. Sb. 30 (72) (1952), 349-462; English transl., Amer. Math. Soc. Transl. (2) 6 (1957), 111-244. MR 13, 904. MR 0047629 (13:904c)
  • [2] N. Jacobson, Structure of rings, Amer. Math. Soc. Colloq. Publ., vol. 37, Amer. Math. Soc., Providence, R. I., 1956. MR 18, 373. MR 0081264 (18:373d)
  • [3] -, Lie algebras, Interscience Tracts in Pure and Appl. Math., no. 10, Interscience, New York, 1962. MR 26 #1345. MR 0143793 (26:1345)
  • [4] B. Kostant, On holonomy and homogeneous spaces, Nagoya Math. J. 12 (1957), 31-54. MR 21 #6003. MR 0107278 (21:6003)
  • [5] O. Loos, Symmetric spaces. II, Benjamin, New York, 1969. MR 39 #365b.
  • [6] K. McCrimmon, On Herstein's theorem relating Jordan and associative algebras, J. Algebra 13 (1969), 382-392. MR 40 #2721. MR 0249476 (40:2721)
  • [7] K. Nomizu, Invariant affine connections on homogeneous spaces, Amer. Math. J. 76 (1954), 33-65. MR 15, 468. MR 0059050 (15:468f)
  • [8] A. Sagle, On simple extended Lie algebras over fields of characteristic zero, Pacific J. Math. 15 (1965), 621-648. MR 32 #7612. MR 0190198 (32:7612)
  • [9] -, Some homogeneous Einstein manifolds, Nagoya Math. J. 39 (1970), 81-106. MR 42 #6748. MR 0271867 (42:6748)
  • [10] A. Sagle and D. Winter, On homogeneous spaces and reductive subalgebras of simple Lie algebras, Trans. Amer. Math. Soc. 128 (1967), 142-147. MR 37 #2910. MR 0227325 (37:2910)
  • [11] A. Sagle and J. Schumi, Multiplications on homogeneous spaces, nonassociative algebras and connections, Pacific J. Math. (to appear). MR 0375165 (51:11361)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1974-0339013-6
Keywords: Reductive pair, homogeneous space, pseudo-Riemannian connection, nonassociative algebra, Jordan algebra, Taylor series
Article copyright: © Copyright 1974 American Mathematical Society

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