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Noncommutative Jordan division algebras


Author: Kevin McCrimmon
Journal: Trans. Amer. Math. Soc. 163 (1972), 215-224
MSC: Primary 17A15
DOI: https://doi.org/10.1090/S0002-9947-1972-0320098-6
MathSciNet review: 0320098
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Abstract: The structure theory for noncommutative Jordan algebras with chain conditions leads to the following simple algebras: (I) division algebras, (II) forms of nodal algebras, (III) algebras of generic degree two, (IV) commutative Jordan matrix algebras, (V) quasi-associative algebras. The chain condition is always satisfied in a division algebra, hence does not serve as a finiteness restriction. Consequently, the general structure of noncommutative Jordan division algebras, even commutative Jordan division algebras, is unknown. In this paper we will classify those non-commutative Jordan division algebras which are forms of algebras of types (II)-(V); this includes in particular all the finite-dimensional ones.


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

  • [1] A. A. Albert, A construction of exceptional Jordan division algebras, Ann. of Math. (2) 67 (1958), 1-28. MR 19, 1036. MR 0091946 (19:1036b)
  • [2] J. Goldman, Nodal algebras of dimension $ {p^3}$, Proc. Amer. Math. Soc. 24 (1970), 156-160. MR 0272842 (42:7723)
  • [3] N. Jacobson, Structure and representations of Jordan algebras, Amer. Math. Soc. Colloq. Publ., vol. 39, Amer. Math. Soc., Providence, R. I., 1968. MR 40 #4330. MR 0251099 (40:4330)
  • [4] L. Kokoris, Nodal non-commutative Jordan rings, Canad. J. Math. 12 (1960), 488-492. MR 22 #6834. MR 0116039 (22:6834)
  • [5] K. McCrimmon, Noncommutative Jordan rings, Trans. Amer. Math. Soc. 158 (1971), 1-33. MR 0310024 (46:9127)
  • [6] -, A note on quasi-associative algebras, Proc. Amer. Math. Soc. 17 (1966), 1455-1459. MR 39 #270. MR 0238910 (39:270)
  • [7] K. McCrimmon and R. D. Schafer, On a class of noncommutative Jordan algebras, Proc. Nat. Acad. Sci. U.S.A. 56 (1966), 1-4. MR 34 #5888. MR 0206063 (34:5888)
  • [8] K. McCrimmon, Quadratic Jordan algebras and cubing operations, Trans. Amer. Math. Soc. 153 (1971), 265-278. MR 0268239 (42:3138)
  • [9] -, Homotopes of noncommutative Jordan algebras, Math. Ann. 191 (1971), 263-270. MR 0313333 (47:1888)
  • [10] -, The Freudenthal-Springer-Tits constructions of exceptional Jordan algebras, Trans. Amer. Math. Soc. 139 (1969), 495-510. MR 39 #276. MR 0238916 (39:276)
  • [11] -, The Freudenthal-Springer-Tits constructions revisited, Trans. Amer. Math. Soc. 148 (1970), 293-314. MR 0271181 (42:6064)
  • [12] R. D. Schafer, An introduction to nonassociative algebras, Pure and Appl. Math., vol. 22, Academic Press, New York, 1966. MR 35 #1643. MR 0210757 (35:1643)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1972-0320098-6
Keywords: Noncommutative Jordan algebra, division algebra
Article copyright: © Copyright 1972 American Mathematical Society

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