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A Kronecker Factorization Theorem for Cayley Algebras and the Exceptional Simple Jordan Algebra

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Part of the Contemporary Mathematicians book series (CM)

Abstract

A well known theorem, due to Wedderburn, states that if \( \mathfrak{B} \) is an arbitrary associative algebra with an identity 1 and \( \mathfrak{A} \) is a finite dimensional central simple subalgebra containing 1, then \( \mathfrak{B} \) is the Kronecker product \( \mathfrak{B} = \mathfrak{A} \otimes \mathfrak{U} \) where \( \mathfrak{U} \) is the subalgebra of elements which commute with every \( a\,\varepsilon \,\mathfrak{A} \). Recently Kaplansky indicated that the same result holds for \( \mathfrak{B} \) any alternative algebra and \( \mathfrak{A} \) an algebra of Cayley numbers.2 It is the purpose of the present note to give a new proof of this result and to show that an analogous result holds for \( \mathfrak{B} \) any Jordan algebra and \( \mathfrak{A} \) an exceptional simple Jordan algebra of finite dimensions.

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References

  1. [1]
    A. A. Albert, “On simple alternative rings,” Canadian Journal of Mathematics, vol. 2 (1952), pp. 129–135.CrossRefGoogle Scholar
  2. [2]
    N. Jacobson, “Structure of alternative and Jordan bimodules,” forthcoming.Google Scholar
  3. [3]
    I. Kaplansky, “Semi-simple alternative rings,” Portugaliae Mathematica, vol. 10 (1951), pp. 37–50.Google Scholar
  4. [4]
    R. D. Schafer, “The exceptional simple Jordan algebras,” American Journal of Mathematics, vol. 70 (1948), pp. 82–84.CrossRefGoogle Scholar
  5. [5]
    T. Nakayama, “Wedderburn’s theorem, weakly normal rings,” Journal of the Mathematical Society of Japan, vol. 5 (1953), pp. 154–170.CrossRefGoogle Scholar

Copyright information

© Birkhäuser Boston 1989

Authors and Affiliations

  1. 1.Yale UniversityUSA

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