Structure of Alternative and Jordan Bimodules

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.

[1]  Max Zorn,et al.  Theorie der alternativen ringe , 1931 .

[2]  Über die Multiplikation quantenmechanischer Größen. II , 1933 .

[3]  A. Albert.  On a Certain Algebra of Quantum Mechanics , 1934 .

[4]  J. Neumann,et al.  On an Algebraic generalization of the quantum mechanical formalism , 1934 .

[5]  R. Moufang,et al.  Zur Struktur von Alternativkörpern , 1935 .

[6]  H. Weyl,et al.  Spinors in n Dimensions , 1935 .

[7]  Garrett Birkhoff,et al.  Representability of Lie Algebras and Lie Groups by Matrices , 1937 .

[8]  E. Witt,et al.  Theorie der quadratischen Formen in beliebigen Körpern. , 1937 .

[9]  E. Witt,et al.  Treue Darstellung Liescher Ringe. , 1937 .

[10]  R. Duffin.  On The Characteristic Matrices of Covariant Systems , 1938 .

[11]  N. Jacobson.  Cayley numbers and normal simple Lie algebras of type $G$ , 1939 .

[12]  N. Jacobson.  Classes of Restricted Lie Algebras of Characteristic p. I , 1941 .

[13]  N. Jacobson.  Classes of restricted Lie algebras of characteristic $p$, II , 1943 .

[14]  I. Kaplansky.  Semi-automorphisms of rings , 1947 .

[15]  A. Albert.  A Structure Theory for Jordan Algebras , 1947 .

[16]  On special Jordan algebras , 1947 .

[17]  German Ancohea.  On Semi-Automorphisms of Division Algebras , 1947 .

[18]  R. D. Schafer.  The Exceptional Simple Jordan Algebras , 1948 .

[19]  Samuel Eilenberg,et al.  Extensions of general algebras , 1948 .

[20]  N. Jacobson.  Isomorphisms of Jordan Rings , 1948 .

[21]  Nathan Jacobson,et al.  Classification and Representation of Semi-Simple Jordan Algebras , 1949 .

[22]  Derivation Algebras and Multiplication Algebras of Semi-Simple Jordan Algebras , 1949 .

[23]  G. Birkhoff,et al.  Representation of Jordan and Lie algebras , 1949 .

[24]  Nathan Jacobson,et al.  Lie and Jordan Triple Systems , 1949 .

[25]  W. Specht.  Gesetze in Ringen. I , 1950 .

[26]  A. Albert.  A theory of power-associative commutative algebras , 1950 .

[27]  A. Albert.  A note on the exceptional Jordan algebra. , 1950, Proceedings of the National Academy of Sciences of the United States of America.

[28]  C Chevalley,et al.  The Exceptional Simple Lie Algebras F(4) and E(6). , 1950, Proceedings of the National Academy of Sciences of the United States of America.

[29]  N. Jacobson,et al.  Jordan homomorphisms of rings , 1950 .

[30]  General Representation Theory of Jordan Algebras , 1951 .

[31]  A THEOREM ON THE REPRESENTATION THEORY OF JORDAN ALGEBRAS , 1951 .

[32]  The Wedderburn principal theorem for Jordan algebras , 1951 .

[33]  R. D. Schafer.  Representations of alternative algebras , 1952 .

[34]  Homomorphisms of Jordan Rings of Self-Adjoint Elements , 1952 .

[35]  On Simple Alternative Rings , 1952, Canadian Journal of Mathematics.