Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Real homogeneous algebras


Author: D. Ž. Djoković
Journal: Proc. Amer. Math. Soc. 41 (1973), 457-462
MSC: Primary 17A99
DOI: https://doi.org/10.1090/S0002-9939-1973-0332902-2
MathSciNet review: 0332902
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ (A,\mu )$ be a finite dimensional real algebra (not necessarily associative) with multiplication $ \mu \ne 0$. Assuming that $ \operatorname{Aut}(A)$ is transitive on one-dimensional subspaces we determine all such algebras. There are up to isomorphism only four such algebras, one in each of the dimensions 1, 3, 6, 7.


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

  • [1] A. Borel, Some remarks about Lie groups transitive on spheres and tori, Bull. Amer. Math. Soc. 55 (1949), 580-587. MR 10, 680. MR 0029915 (10:680c)
  • [2] -, Le plan projectif des octaves et les sphères comme espaces homogènes, C.R. Acad. Sci. Paris 230 (1950), 1378-1380. MR 11, 640. MR 0034768 (11:640c)
  • [3] N. Bourbaki, Eléments de mathématique. Algèbre. Chaps. 1-3, Hermann, Paris, 1970. MR 43 #2.
  • [4] F. Gross, Finite automorphic algebras over GF(2), Proc. Amer. Math. Soc. 31 (1972), 10-14. MR 0286856 (44:4063)
  • [5] G. Hochschild, Introduction to affine algebraic groups, Holden-Day, San Francisco, Calif., 1971. MR 43 #3268. MR 0277535 (43:3268)
  • [6] N. Jacobson, Lie algebras, Interscience Tracts in Pure and Appl. Math., no. 10, Interscience, New York, 1962. MR 26 #1345. MR 0143793 (26:1345)
  • [7] A. I. Kostrikin, On homogeneous algebras, Izv. Akad. Nauk SSSR Ser. Mat. 29 (1965), 471-484; English transl., Amer. Math. Soc. Transl. (2) 66 (1968), 130-144. MR 31 #219. MR 0175943 (31:219)
  • [8] D. Montgomery and H. Samelson, Transformation groups on spheres, Ann. of Math. (2) 44 (1943), 454-470. MR 5, 60. MR 0008817 (5:60b)
  • [9] D. Montgomery, Simply connected homogeneous spaces, Proc. Amer. Math. Soc. 1 (1950), 467-469. MR 12, 242. MR 0037311 (12:242c)
  • [10] J. Poncet, Groupes de Lie compacts de transformations de l'espace euclidien et les sphères comme espaces homogènes, Comment. Math. Helv. 33 (1959), 109-120. MR 21 #2708. MR 0103946 (21:2708)
  • [11] E. E. Shult, On the triviality of finite automorphic algebras, Illinois J. Math. 13 (1969), 654-659. MR 40 #1442. MR 0248188 (40:1442)
  • [12] L. Sweet, On homogeneous algebras, Ph.D. Thesis, University of Waterloo, 1973.
  • [13] S. Świerczkowski, Homogeneous Lie algebras, Bull. Austral. Math. Soc. 4 (1971), 349-353. MR 43 #6277. MR 0280557 (43:6277)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 17A99

Retrieve articles in all journals with MSC: 17A99


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1973-0332902-2
Keywords: Homogeneous algebra, Lie group, Lie algebra, linear representation, symmetric and skew-symmetric tensors, weights
Article copyright: © Copyright 1973 American Mathematical Society

American Mathematical Society