Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

A note on quadratic Jordan algebras of degree $ 3$


Author: M. L. Racine
Journal: Trans. Amer. Math. Soc. 164 (1972), 93-103
MSC: Primary 17A15
DOI: https://doi.org/10.1090/S0002-9947-1972-0304447-0
MathSciNet review: 0304447
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: McCrimmon has defined a class of quadratic Jordan algebras of degree 3 obtained from a cubic form, a quadratic mapping and a base point. The structure of such an algebra containing no absolute zero-divisor is determined directly. A simple proof of Springer's Theorem on isomorphism of reduced simple exceptional quadratic Jordan algebras is given.


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

  • [1] F. van der Blij and T. A. Springer, The arithmetics of octaves and of the group $ {G_2}$, Nederl. Akad. Wetensch. Proc. Ser. A 62=Indag. Math. 21 (1959), 406-418. MR 27 #2533. MR 0152555 (27:2533)
  • [2] J. Faulkner, Octonion planes defined by quadratic Jordan algebras, Mem. Amer. Math. Soc. No. 104 (1970). MR 0271180 (42:6063)
  • [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] -, Structure theory of quadratic Jordan algebras, Lecture Notes, Tata Institute, Bombay, 1970.
  • [5] N. Jacobson and K. McCrimmon, Quadratic Jordan algebras of quadratic forms with base points, Indian J. (to appear). MR 0357531 (50:9999)
  • [6] K. McCrimmon, The Freudenthal-Springer-Tits constructions of exceptional Jordan algebras, Trans. Amer. Math. Soc. 139 (1969), 495-510. MR 39 #276. MR 0238916 (39:276)
  • [7] -, A note on reduced Jordan algebras, Proc. Amer. Math. Soc. 19 (1968), 964-970. MR 37 #2826. MR 0227241 (37:2826)
  • [8] T. A. Springer, On a class of Jordan algebras, Nederl. Akad. Wetensch. Proc. Ser. A 62 =Indag. Math. 21 (1959), 254-264. MR 22 #1607. MR 0110739 (22:1607)
  • [9] -, The classification of reduced exceptional simple Jordan algebras, Nederl. Akad. Wetensch. Proc. Ser. A 63=Indag. Math. 22 (1960), 414-422. MR 26 #5035. MR 0147520 (26:5035)
  • [10] T. Tamagawa, Mimeographed notes on quadratic forms, Yale University, New Haven, Conn., 1969.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 17A15

Retrieve articles in all journals with MSC: 17A15


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1972-0304447-0
Keywords: Quadratic Jordan algebras, cubic form, isomorphism, exceptional Jordan algebra
Article copyright: © Copyright 1972 American Mathematical Society

American Mathematical Society