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

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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.

The results of this paper were obtained while the author held a Guggenheim Memorial Fellowship.

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References

  1. A. A. Albert, “On simple alternative rings,” Canadian Journal of Mathematics, vol. 2 (1952), pp. 129–135.

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  2. N. Jacobson, “Structure of alternative and Jordan bimodules,” forthcoming.

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  3. I. Kaplansky, “Semi-simple alternative rings,” Portugaliae Mathematica, vol. 10 (1951), pp. 37–50.

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  4. R. D. Schafer, “The exceptional simple Jordan algebras,” American Journal of Mathematics, vol. 70 (1948), pp. 82–84.

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  5. T. Nakayama, “Wedderburn’s theorem, weakly normal rings,” Journal of the Mathematical Society of Japan, vol. 5 (1953), pp. 154–170.

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© 1989 Birkhäuser Boston

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Jacobson, N. (1989). A Kronecker Factorization Theorem for Cayley Algebras and the Exceptional Simple Jordan Algebra. In: Nathan Jacobson Collected Mathematical Papers. Contemporary Mathematicians. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-3694-8_14

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  • DOI: https://doi.org/10.1007/978-1-4612-3694-8_14

  • Publisher Name: Birkhäuser Boston

  • Print ISBN: 978-1-4612-8215-0

  • Online ISBN: 978-1-4612-3694-8

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