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Transactions of the American Mathematical Society

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Octonion planes over local rings

Author: Robert Bix
Journal: Trans. Amer. Math. Soc. 261 (1980), 417-438
MSC: Primary 17C40; Secondary 20H25
MathSciNet review: 580896
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Abstract: Let $ \mathcal{D}$ be an octonion algebra which is a free module over a local ring R and let $ J = H({\mathcal{D}_3},\gamma )$ be the quadratic Jordan algebra of Hermitian 3-by-3 matrices over R. We define the octonion plane determined by J and prove that every collineation is induced by a norm semisimilarity of J. We classify the subgroups of the collineation group normalized by the little projective group.

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  • [1] R. Bix, Separable Jordan algebras over commutative rings, Dissertation, Yale University, 1977.
  • [2] -, Separable Jordan algebras over commutative rings. I, J. Algebra 57 (1979), 111-143. MR 533104 (80i:17018)
  • [3] C. Chevalley, The algebraic theory of spinors, Columbia Univ. Press, New York, 1954. MR 0060497 (15:678d)
  • [4] F. Demeyer and E. Ingraham, Separable algebras over commutative rings, Lecture Notes in Math., vol. 181, Springer-Verlag, New York, 1971. MR 0280479 (43:6199)
  • [5] J. Faulkner, Octonion planes defined by quadratic Jordan algebras, Mem. Amer. Math. Soc., No. 104, 1970. MR 0271180 (42:6063)
  • [6] I. Herstein, Noncommutative rings, Carus Math. Mono., No. 15, Math. Assoc. Amer., Washington, D. C, 1968. MR 0227205 (37:2790)
  • [7] N. Jacobson, Structure and representations of Jordan algebra, Amer. Math. Soc. Colloq. Publ., vol. 39, Amer. Math. Soc., Providence, R. I., 1968. MR 0251099 (40:4330)
  • [8] D. James, On the structure of orthogonal groups over local rings, Amer. J. Math. 95 (1973), 255-265. MR 0330316 (48:8653)
  • [9] W. Klingenberg, Orthogonale Gruppen über lokalen Ringen, Amer. J. Math. 83 (1961), 281-320. MR 0124414 (23:A1726)
  • [10] K. McCrimmon, The Freudenthal-Springer-Tits construction of exceptional Jordan algebras, Trans. Amer. Math. Soc. 139 (1969), 495-510. MR 0238916 (39:276)
  • [11] B. McDonald, Geometrie algebra over local rings, Dekker, New York, 1976. MR 0476639 (57:16198)

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Keywords: Octonion planes, exceptional Jordan algebras, norm semisimilarities, orthogonal groups over local rings
Article copyright: © Copyright 1980 American Mathematical Society

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