Structure of Alternative and Jordan Bimodules

Part of the Contemporary Mathematicians book series (CM)


The notion of a bimodule for a class of algebras defined by multilinear identities has been introduced by Eilenberg [13]. If \( \mathfrak{A} \) is in the class of associative algebras or in the class of Lie algebras, then this notion is the familiar one for which we are in possession of well-worked theories. The study of bimodules (or representations) of Jordan algebras was initiated by the author in a recent paper [21]. Subsequently the alternative case was considered by Schafer [32]. In our paper we introduced the basic concepts of the Jordan theory and we proved complete reducibility of the bimodules and the analogue of Whitehead’s first lemma for finite dimensional semi-simple Jordan algebras of characteristic 0. Similar results on alternative algebras, based on those in the Jordan case, were obtained by Schafer. The principal tool in our paper was the notion of a Lie triple system. This permitted the application of important results on the structure and representation of Lie algebras to the problems on Jordan and alternative algebras. This method has one nice feature, namely, it is a general one which does not require a consideration of cases.


Associative Algebra Clifford Algebra Jordan Algebra Base Field Central Simple Algebra 
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  1. 1.
    A. A. Albert, On a certain algebra of quantum mechanics, Ann. of Math., 35 (1934), 65–73.CrossRefGoogle Scholar
  2. 2.
    A. A. Albert, A structure theory for Jordan algebras, Ann. of Math., 48 (1947), 446–467.Google Scholar
  3. 3.
    A. A. Albert, A note on the exceptional Jordan algebra, Proc. Nat. Acad. Sci., 36 (1950), 372–374.CrossRefGoogle Scholar
  4. 4.
    A. A. Albert, A theory of power associative commutative algebras, Trans. Amer. Math Soc., 69 (1950), 503–527.Google Scholar
  5. 5.
    A. A. Albert, On simple alternative rings, Can. J. of Math., 4 (1952), 129–135.CrossRefGoogle Scholar
  6. 6.
    A. S. Amitsur, On the identities of PI rings, Proc. Amer. Math. Soc., 4, (1953), 27–35.Google Scholar
  7. 7.
    G. Ancochea, On semi-automorphisms of division algebras, Ann. of Math., 48 (1947), 147–154.CrossRefGoogle Scholar
  8. 8.
    G. Birkhoff, Representability of Lie algebras, etc.. Ann. of Math., 38 (1937), 326–332.Google Scholar
  9. 9.
    G. Birkhoff and P. Whitman, Representations of Jordan and Lie algebras, Trans. Amer. Math. Soc., 65 (1949), 116–136.CrossRefGoogle Scholar
  10. 10.
    R. Brauer and H. Weyl, Spinors in n dimensions, Amer. J. of Math., 57 (1935), 425–449.CrossRefGoogle Scholar
  11. 11.
    C. Cheyalley and R. D. Schafer, The exceptional simple Lie algebras \( {\mathfrak{F}_4} \) and \( {\mathfrak{E}_6} \), Proc. Nat. Acad. Sci., 36 (1950), 137–141.CrossRefGoogle Scholar
  12. 12.
    R. J. Duffin, On the characteristic matrices of covariant systems, Phys. Rev., 54 (1938), 1114.CrossRefGoogle Scholar
  13. 13.
    S. Eilenberg, Extensions of general algebras, Annales de la Société Polonaise de Mathématique, 21 (1948), 125–34.Google Scholar
  14. 14.
    H. Freudenthal, Oktaven, Ausnahmengruppen und Oktavengeometrie, Utrecht 1951.Google Scholar
  15. 15.
    F. O. Jacobson and N. Jacobson, Classification and representation of semi-simple Jordan algebras, Trans. Amer. Math. Soc. 65 (1949), 141–169.Google Scholar
  16. 16.
    N. Jacobson, Cayley numbers and simple Lie algebras of type G, Duke Math. J., 5 (1939), 775–783.CrossRefGoogle Scholar
  17. 17.
    N. Jacobson, Classes of restricted Lie algebras of characteristic p, I, Amer. J. of Math. 63 (1941), 481–515.CrossRefGoogle Scholar
  18. 18.
    N. Jacobson, Isomorphisms of Jordan rings, Amer. J. of Math., 70 (1948), 317–326.CrossRefGoogle Scholar
  19. 19.
    N. Jacobson, Derivation algebras and multiplication algebras of semi-simple Jordan algebras, Ann. of Math., 50 (1949), 866–874.CrossRefGoogle Scholar
  20. 20.
    N. Jacobson, Lie and Jordan triple systems, Amer. J. of Math., 17 (1949), 149–170.CrossRefGoogle Scholar
  21. 21.
    N. Jacobson, General representation theory of Jordan algebras, Trans. Amer. Math. Soc., 70 (1951), 509–530.CrossRefGoogle Scholar
  22. 22.
    N. Jacobson and C. E. Rickart, Jordan Homomorphism of Rings, Trans. Amer. Math. Soc., 69 (1950), 479–502.Google Scholar
  23. 23.
    N. Jacobson and C. E. Rickart, Homomorphisms of Jordan rings of selfadjoint elements. Trans. Amer. Math. Soc., 72 (1952), 310–322.CrossRefGoogle Scholar
  24. 24.
    P. Jordan, Über die Multiplikation quanten-mechanischer Grössen, Zeitschrift für Physik, 80 (1933), 285–291.CrossRefGoogle Scholar
  25. 25.
    P. Jordan, J. von Neumann and E. Wigner, On an algebraic generalization of the quantum mechanical formalism, Ann. of Math., 35 (1934), 29–64.CrossRefGoogle Scholar
  26. 26.
    G. K. Kalisch, On special Jordan algebras, Trans. Amer. Math. Soc., 61 (1947), 482–494.CrossRefGoogle Scholar
  27. 27.
    I. Kaplansky, Semi-automorphisms of rings, Duke Math. J. 14 (1947), 521–527.CrossRefGoogle Scholar
  28. 28.
    W. H. Mills, A theorem on the represention theory of Jordan algebras, Pacific J. of Math., 1 (1951), 255–264.Google Scholar
  29. 29.
    R. Moufang, Zur Struktur von alternativen Körpern, Math. Anna. 110 (1937), 416–430.CrossRefGoogle Scholar
  30. 30.
    A.J. Penico, The Wedderburn principal theorem for Jordan algebras, Trans. Amer. Math. Soc., 70 (1951), 404–421.CrossRefGoogle Scholar
  31. 31.
    R. D. Schafer, The exceptional simple Jordan algebras, Amer. J. of Math., 70 (1948), 82–94.CrossRefGoogle Scholar
  32. 32.
    R. D. Schafer, Representations of alternative algebras, Trans. Amer. Math. Soc., 72 (1952), 1–17.CrossRefGoogle Scholar
  33. 33.
    W. Specht, Gesetze in Ringen I, Math Zeitschr. 52, (1950), 557–589.CrossRefGoogle Scholar
  34. 34.
    N. Svartholm, On the algebras of relativistic quantum mechanics, Proc. of the Royal Physiographical Soc. of Lond, 12 (1942), 94–108.Google Scholar
  35. 35.
    E. Witt, Theorie der quadratischen Formen in beliebigen Körpern, Journal für die Reine und Angewandete Mathematik, 176 (1937), 31–44.CrossRefGoogle Scholar
  36. 36.
    E. Witt, Treue Darstellung Liescher Ringe, J. Reine Angew. Math., 177 (1937) 152–160.CrossRefGoogle Scholar
  37. 37.
    M. Zorn, Theorie der Alternativen Ringe, Abh. Mat. Semi. Hamburg. Univ. 8 (1930), 123–147.CrossRefGoogle Scholar

Copyright information

© Birkhäuser Boston 1989

Authors and Affiliations

  1. 1.Yale UniversityUSA

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